снартев 30
ANISOTROPIC AND INHOMOGENEOUS COSMOLOGIES

§30.1. WHY IS THE UNIVERSE SO HOMOGENEOUS AND ISOTROPIC?

This chapter is entirely Track 2.
The main text requires no special preparation, although Chapters 27-29 would be helpful.
Box 30.1 contains more technical sections: ideal preparation for it would be Chapters 4,9-14, 21, and 27-29, plus §25.2; minimal preparation would be exercises

9.13, 9.14

, and 25.2, Chapter 21 through §21.8, and §§

27.8, 27.11

, and 29.2.
Chapter 30 is not needed as preparation for any later chapter.

Motivation for studying inhomogeneous and anisotropic cosmologies: Why is universe so uniform?

The last three chapters studied the Friedmann cosmological models and the relatively satisfactory picture they give of the universe and its evolution. This chapter describes less simplified cosmological models, and uses them to begin answering the question, "Why are the very simple Friedmann models satisfactory?" This question is intended to probe more deeply than the first, obvious answer-namely, that the models are satisfactory because they do not contradict observations. Accepting the agreement with observations, we want to understand why the laws of physics should demand (rather than merely permit) a universe that is homogeneous and isotropic to high accuracy on large scales. Because this question cannot be answered definitively in 1972, many readers will prefer to omit this chapter on the first reading and return to it only after they have surveyed the major results in other areas such as black holes (Chapter 33), gravitational waves (Chapters 35-37), and solar-system experiments (Chapter 40).
The approach described here to the question "Why is the universe so highly symmetric?" is to ask Einstein's equations to describe what would have happened if the universe had started out highly irregular.
The first step in this approach is to ask what would have happened if the universe had started a little bit irregular. This problem can be tackled by analyzing small perturbations away from the high symmetry of the Friedmann models. Such an analysis is most fruitful in its discussion of the beginnings of galaxy formation, and
in its ability to relate small upper limits on the present-day anisotropy of the microwave background radiation to limits on density and temperature irregularities that might have existed ten billion years ago, when the radiation was emitted. These studies are described so well in the book by Zel'dovich and Novikov (1974) [see also Field (1973), Peebles (1969), Peebles and Yu (1970), Jones and Peebles (1972), and references cited therein] that we omit them here.

Another approach is to allow large deviations from the symmetry of the Friedmann universes, but to put the asymmetries into only a few degrees of freedom.

§30.2. THE KASNER MODEL FOR AN ANISOTROPIC UNIVERSE

The prototype for cosmological models with great asymmetry in a few degrees of freedom is the Kasner (1921a) metric,

\begin{matrix} (30.1) & d s^{2} = - d t^{2} + t^{2 p_{1}} d x^{2} + t^{2 p_{2}} d y^{2} + t^{2 p_{3}} d z^{2} \end{matrix}

which was first studied as a cosmological model by Schücking and Heckmann (1958). In this metric the

p_{i}

are constants satisfying

\begin{matrix} (30.2) & p_{1} + p_{2} + p_{3} = {(p_{1})}^{2} + {(p_{2})}^{2} + {(p_{3})}^{2} = 1 . \end{matrix}

Each

t =

constant hypersurface of this cosmological model is a flat three-dimensional space. The world lines of constant

x, y, z

are timelike geodesics along which galaxies or other matter, treated as test particles, can be imagined to move. This model represents an expanding universe, since the volume element

\sqrt{- g} = \sqrt{(3) g} = t

is constantly increasing. But it is an anisotropically expanding universe. The separation between two standard (constant

x, y, z

) observers is

t^{p_{1}} Δ x

if only their

x

-coordinates differ. Thus, distances parallel to the

x

-axis expand at one rate,

ℓ_{1} \propto t^{p_{1}}

, while those along the

y

-axis can expand at a different rate,

ℓ_{2} \propto t^{p_{2}}

. Most remarkable perhaps is the fact that along one of the axes distances contract rather than expand. This contraction shows up mathematically in the fact that equations (30.2) require one of the

p

's, say

p_{1}

, to be nonpositive:

\begin{matrix} (30.3) & - \frac{1}{3} \leq p_{1} \leq 0 . \end{matrix}

As a consequence, in a universe of this sort, if black-body radiation were emitted at one time

t

and never subsequently scattered, later observers would see blue shifts near one pair of antipodes on the sky and red shifts in most other directions. In terms of this example, the fundamental cosmological question is why the Friedmann metrics should be a more accurate approximation to the real universe than is this Kasner metric.

Kasner metric: an example of an anisotropic model universe

§30.3. ADIABATIC COOLING OF ANISOTROPY

In seeking an answer, ask a question. Ask, in particular, what would become of a universe that starts out near

t = 0

with a form described by the Kasner metric of equation (30.1). This metric is an exact solution of the vacuum Einstein equation

G = 0

. It approximates a situation where the matter terms in the Einstein equations are negligible by comparison with typical non-zero components of the Riemann tensor. Schücking and Heckmann (1958) give solutions with matter included as a pressureless fluid. In this situation, the curvature of empty spacetime dominates both the geometry and the expansion rate at early times,

t ⟶ 0

; but after some characteristic time

t_{m}

the matter terms become more important, and the metric reduces asymptotically to the homogeneous, isotropic model with

k = 0

This example illustrates the possibility that the universe might achieve a measure of isotropy and homogeneity in old age, even if it were born in a highly irregular state. Whether the symmetry of our universe can be explained along these lines is not yet clear in 1972. The model universe just mentioned is only a hint, especially since the critical parameter

t_{m}

can be given any value whatsoever.

The standard Einstein general-relativity physics of this model can be described in other language (Misner, 1968) by ascribing to the anisotropic motions of empty

Anisotropy energy

Adiabatic cooling of anisotropy spacetime an "effective energy density"

ρ_{aniso}

, which enters the

G_{00}

component of the Einstein equation on an equal footing with the matter-energy density, and thereby helps to account for the expansion of the universe:

\begin{matrix} (30.4) & H^{2} = {(\frac{1}{3} \frac{d}{d t} \ln \sqrt{(3)} g)}^{2} = \frac{8 π}{3} (ρ_{aniso} + ρ_{matter}) . \end{matrix}

The anisotropy energy density is found to have an equation of state

ρ_{aniso} \propto^{(3)} g^{- 1} = (volume)^{- 2},

while

ρ_{matter} \propto^{(3)} g^{- γ / 2} = (volume)^{- γ} .

For pressureless matter

γ = 1

; for a radiation fluid

γ = 4 / 3

; for a nonrelativistic ideal gas

γ = 5 / 3

This arrangement of the Einstein equation allows one to think of the anisotropy motions as being adiabatically cooled by the expansion of the universe, just as the thermal motions of an ideal gas would be. Since the adiabatic index for homogeneous anisotropy is

γ = 2

, the anisotropy will be the dominant source of "effective energy" in a highly compressed state, whereas the matter will dominate in an expanded state.

§30.4. VISCOUS DISSIPATION OF ANISTROPY

The model universe sketched above can be further elaborated by introducing dissipative mechanisms that convert anisotropy energy into thermal energy. Suppose that
such an anisotropic universe were filled at one time with thermal radiation. If the radiation were collisionless or nearly so, the quanta moving parallel to the contracting

x

-axis would get blueshifted and would develop an energy distribution corresponding to a high temperature. The quanta moving parallel to the other (expanding) axes would be redshifted to an energy distribution corresponding to a low temperature. Any collisions taking place between these two systems of particles would introduce a "thermal contact" between them, and would transfer energy from the hot system to the cold one, with a corresponding large production of entropy. This provides an irreversible dissipative process, which decreases

ρ_{aniso}

and increases

ρ_{radiation}

relative to the values they would have had under conditions of adiabatic expansion. [For further details, see, e.g., Matzner and Misner (1972).]

It is possible that both the adiabatic cooling of anisotropy and the dissipation of anisotropy by its action on a gas of almost collisionless quanta have played significant roles in the evolution of our universe. In particular, neutrinos above

10^{10} K

may have undergone sufficient

ν

-e scattering to have provided strong dissipation during the first few seconds of the life of the universe.

§30.5. PARTICLE CREATION IN AN ANISOTROPIC UNIVERSE

Adiabatic cooling and viscous dissipation might not be the chief destroyers of anisotropy in an expanding universe. More powerful still might be another highly dissipative process, which might occur at still earlier times, very near the initial "singularity." This is a process of particle creation which was first treated by DeWitt (1953), then explored by Parker (1966 and 1969) for isotropic cosmologies and finally by Zel'dovich (1970) in the present context of anisotropic cosmologies. In this process one again turns to the Kasner metric for the simplest example, but now quantummechanical considerations enter the picture. One realizes that not only would real quanta propagating in different directions be subject to red shifts and blue shifts, but that virtual quanta must be considered as well. Vacuum fluctuations (zero-point oscillations) entail a certain minimum number of virtual quanta, which are subject to the redshifting and blueshifting action of the strong gravitational fields. Virtual quanta that are blueshifted sufficiently violently can materialize as real particles, thanks to their energy gain. In this context "sufficiently violently" means not adiabatically.

In an adiabatic expansion, the number of particles does not change, although the energy of each one does. This adiabatic limit is just the geometric-optics approximation to wave equations, which was discussed in

§ 22.5

. There one saw that, if spacetime were not flat on the scale of a wavelength, then the wave equation could not be replaced by a particle description with conserved particle numbers. Thus, the adiabatic limit (geometric-optics approximation) is violated in the conditions of high curvature near the singularity at the beginning of the universe.

By studying wave equations in the Kasner background metric, Zel'dovich and Starobinsky (1971) find quantitatively the consequences of the failure of the adia-

Creation of particles by anisotropy of expansion

Anisotropy might have created the matter content of our universe, damping itself out in the process

Inhomogeneous cosmological models:
(1) with spherical symmetry
(2) with (rather symmetric) gravitational waves
(3) near a singularity, with few or no symmetries
batic approximation near the singularity. Classically, the amplitudes of waves at frequencies comparable to the Hubble constant for any given epoch increase faster than a simple blue-shift calculation would imply (amplification through parametric resonance). Quantum-mechanically, the same amplification, applied to zero-point oscillations, leads to the creation of particle-antiparticle pairs. The calculations indicate that this effect is very strong at the characteristic time

t_{q} = \sqrt{G ℏ / c^{5}} ≃ 10^{- 43}

sec. (All calculations performed thus far are inadequate when the effect becomes strong, thus for

t ≦ t_{q}

For the creation of massless particles, it is essential that an anisotropically expanding universe be postulated (except for scalar particles, for which particle creation occurs already in the Friedmann universe, unless the particle satisfies the conformalinvariant wave equation). The isotropic Friedmann universes are all conformally flat, so that solutions of the wave equation for a field of zero rest mass can be given in terms of solutions for flat-space wave equations where there is no particle creation. There is some particle creation even in the isotropic Friedmann universe when the particle has finite rest mass and low energy. However, the particle-creation process normally uses anisotropy energy as the energy supply that it converts into radiation energy.
The pioneering work by Parker and Zel'dovich suggests that one should study in detail cosmological models in which the initial conditions are a singularity, and in which quantum effects near the time

t = t_{q}

dissipate all anisotropies and simultaneously give rise to the matter content of the model. This program of research, which is in its infancy, seems to require extrapolating laws of physics down to the very natural looking but preposterously small dimension

\sqrt{G ℏ / c^{5}} ≃ 10^{- 43}

sec, or equivalently

\sqrt{G ℏ / c^{3}} \sim 10^{- 33} cm

§30.6. INHOMOGENEOUS COSMOLOGIES

The model universes considered above were all homogeneous although anisotropic. It is also crucial to study inhomogeneous cosmological models, in which the metric has a nontrivial dependence on the space coordinates. One class of such models is spherically symmetric universes, where the matter density, expansion rate, and all other locally measurable physical quantities have spherical symmetry about some preferred origin. Models of this sort were first considered by Lemaitre (1933a,b), Tolman (1934b), and Datt (1938), and were also treated by Bondi in 1947. These models provide a means for studying density perturbations of large amplitude.

A recent tool is making it possible to study large-amplitude, spatially varying curvature perturbations of other symmetries; this tool is the Gowdy (1971, 1973) metrics. These metrics, which are exact solutions of the Einstein equations, represent closed universes with various topologies (

S^{3}, S^{1} \times S^{2}, T^{3}

) containing gravitational waves. The wave form in these solutions is essentially arbitrary, but all the waves propagate along a single preferred direction and have a common polarization.

A rather different approach to understanding the behavior of inhomogeneous and anisotropic solutions of the Einstein equations has been developed by Khalatnikov,

Lifshitz, and their colleagues. Rather than truncate the Einstein theory by limiting attention to specialized situations where exact solutions can be obtained, they have sought to study the widest possible class of solutions, but to describe their behavior only in the immediate neighborhood of the singularity. These studies give a greatly enhanced significance to some of the exact solutions, by showing that phenomena found in them are in fact typical of much broader classes of solutions.

Thus, in the first large class of solutions studied [Lifshitz and Khalatnikov (1963)], it was found that near the singularity solutions containing matter showed no features not already found in the vacuum solutions. Furthermore, space derivatives in the Einstein equations became negligible near the singularity in these solutions, with the consequence that a metric of the Kasner form [equation (30.1)] described the local behavior of spacetime near the singularity, but with a different set of

p_{i}

values possible at each point of the singular hypersurface. Subsequently, broadened studies of solutions near a singularity [Belinsky and Khalatnikov (1970)] showed that the mixmaster universe [Misner (1969b); Belinsky, Khalatnikov, and Lifshitz (1970)] is a still better homogeneous prototype for singularity behavior than the Kasner metric.

§30.7. THE MIXMASTER UNIVERSE

The simplest example of a mixmaster universe is described in Box 30.1. It shows how, near the singularity, the Kasner exponents

p_{i}

can become functions of time. The result is most simply described in terms of the Khalatnikov-Lifshitz parameter

u

\begin{aligned} p_{1} & = - u / (1 + u + u^{2}), \\ (30.5) & p_{2} & = (1 + u) / (1 + u + u^{2}), \\ p_{3} & = u (1 + u) / (1 + u + u^{2}) . \end{aligned}

As one extrapolates backward in time toward the singularity, one finds that the expansion rates in the three principal directions correspond to those of the Kasner metric of equation (30.1), with

p_{i}

values corresponding to some fixed

u

parameter. In these mixmaster models, however, the metric is not independent of the space coordinates (the spacelike hypersurfaces can, for instance, have the same 3 -sphere topology as the closed Friedmann universes).

The Kasner-like behavior at fixed

u

can persist through many decades of volume expansion before effects of the spatial derivatives of the metric come into play. The role then played by the space curvature is brief and decisive. The expansion is converted from a type corresponding to a parameter value

u = u_{0}

to a type corresponding to the value

u = - u_{0}

(which is equivalent, under a relabeling of the axes, to the value

u = u_{0} - 1

). Extrapolating still farther back toward the singularity, one finds a previous period with

u = u_{0} - 2

. Throughout an entire sequence

u = u_{0}

u_{0} - 1, u_{0} - 2, u_{0} - 3, \dots

, with

u_{0} ≫ 1

, nearly the entire volume expansion is due to expansion in the 3 -direction, whereas the 1 - and 2 -directions change very little, alternating at each step between expansion and contraction. Sufficiently far in the past, however, such a sequence leads to a value of

u

between 0 and 1 . This value

Mixmaster universe:
(1) "anisotropy oscillations" explained in terms of Kasner model
(2) as a prototype for generic behavior near singularities

Are there any other generic types of behavior near singularities?
can be interpreted as the starting point for another, similar sequence, through the transformation

u ⟶ 1 / u

, which interchanges the names of axes 2 and 3 .

The extrapolation of the universe's evolution back toward the singularity at

t = 0

therefore shows an extraordinarily complex behavior, in which similar but not precisely identical sequences of behavior are repeated infinitely many times. In terms of a time variable which is approximately

\log (\log t^{- 1})

, these behaviors are quasiperiodic. In the generic example to which the Khalatnikov-Lifshitz methods lead, one has a metric whose asymptotic behavior near the singularity is at each point of the singular hypersurface described by a mixmaster-type behavior, but with the principal axes of expansion changing their directions as well as their roles (as characterized by the

u

parameter) at each step, and with the mixmaster parameters spatially variable. [For more details see Belinsky, Lifshitz, and Khalatnikov (1971), and Ryan (1971, 1972).]

It is not yet (1972) known whether there are important solutions or classes of solutions relevant to the cosmological problem, with asymptotic singularity behavior not described by the Khalatnikov-Lifshitz generic case. The difficulty in reaching a definitive assessment here is that Khalatnikov and Lifshitz use essentially local methods, confined to a single coordinate patch, whereas the desired assessment poses an essentially global question. The global approaches (described in Chapter 34) have not, however, provided any comparable description of the nature of the singularity whose necessity they prove. One attempt to bridge these differences in technique and content is the work by Eardley, Liang, and Sachs (1972).
(continued on page 815)

Box 30.1 THE MIXMASTER COSMOLOGY

The Mixmaster Cosmology is a valuable example. As described in

§ 30.7

, it shows a singularity behavior which illustrates most of the features of the most general examples known. In particular, it shows how properties of empty space reminiscent of an elastic solid become evident near the cosmological singularity.

The mathematical path to this example, as given in this box, also illustrates several important techniques in using the variational principles for the Einstein equations to elucidate the solution of these equations. The Mixmaster example can also be used to provide simple examples of superspace ideas and of quantum formulations of the laws of gravity [Misner (1972a)].

A Generalized Kasner Model

Two generalizations must be implemented in order to progress from the Kasner example (30.1) of a cosmological singularity to the Mixmaster example. The first is to allow a more general timedependence while preserving some of the simplicity of the conditions (30.2) on the exponents

p_{i}

. Note that these exponents satisfy, e.g.,

p_{2} \equiv

d \ln g_{22} / d \ln g

. Therefore one is led to parametrize the

3 \times 3

spatial metric as

\begin{matrix} (1) & g_{i j} = e^{2 α} {(e^{2 β})}_{i j} \end{matrix}

or equivalently,

(\ln g)_{i j} = 2 α δ_{i j} + 2 β_{i j}

, where

β_{i j}

is a traceless

3 \times 3

symmetric matrix, and the exponential is a matrix power series, so

\det e^{2 β} = 1

and

\begin{matrix} (2) & \sqrt{g} = e^{3 α} \end{matrix}

For the purposes of this paragraph only, define

p_{i j} = d (\ln g)_{i j} / d \ln \det g

. Then from equations (1) and (2), one computes

\begin{matrix} (3) & p_{i j} = \frac{1}{3} [δ_{i j} + (d β_{i j} / d α)]; \end{matrix}

so the one Kasner condition

1 = \sum_{i} p_{i} \equiv trace p_{i j} = 1 + \frac{1}{3} trace (d β / d α)

is an identity in view of trace

β_{i j} = 0

. The second condition on the Kasner exponents is trace

(p^{2}) =

1 , and becomes

{(d β_{i j} / d α)}^{2} = 6

by equation (3). This is not an identity, but a consequence of the Einstein equations in empty space. For the (Bianchi Type I) metric

\begin{matrix} (4) & d s^{2} = - d t^{2} + e^{2 α} {(e^{2 β})}_{i j} d x^{i} d x^{j} \end{matrix}

and in the case when

β_{i j}

is diagonal, the Einstein equations are,

\begin{matrix} (5) & {(\frac{d α}{d t})}^{2} = \frac{8 π}{3} [T^{00} + \frac{1}{16 π} {(d β_{i j} / d t)}^{2}] \end{matrix}

and

\begin{matrix} (6) & e^{- 3 α} \frac{d}{d t} (e^{3 α} \frac{d β_{i j}}{d t}) = 8 π (T_{i j} - \frac{1}{3} δ_{i j} T_{k k}), \end{matrix}

together with a redundant equation involving

T_{k k}

and the equation

T_{0 k} = 0

. [The stress components here refer to an orthonormal frame with basis 1 -forms

ω^{i} = e^{α} {(e^{β})}_{i j} d x^{j}

.] From equation (5) one immediately derives

\begin{matrix} (7) & ρ_{aniso(t)} = (c^{2} / 16 π G) {(d β_{i j} / d t)}^{2} \end{matrix}

as a formula for the effectiveness of Type I anisotropy in contributing to the Hubble constant

H =

d α / d t

on a basis comparable to matter energy, as in equation (30.4). Similarly, for equation (6) in the case of fluid matter (isotropic pressures), the stress terms vanish, and one obtains

ρ_{aniso(I)} e^{6 α} =

const., as in the equation following (30.4). The Kasner condition

Σ p_{i}^{2} = 1

{(d β_{i j} / d α)}^{2} = 6

follows from equation (5) whenever

T^{00} ≪ ρ_{aniso}

In the diagonal case,

β_{i j}

has only two independ-
ent components, and it is convenient at times to define them explicitly by the parameterization

\begin{aligned} β_{11} = β_{+} + \sqrt{3} β_{-}, \\ (8) & β_{22} = β_{+} - \sqrt{3} β_{-}, \\ β_{33} = - 2 β_{+} . \end{aligned}

For these the Kasner condition

{(d β_{i j} / d α)}^{2} = 6

becomes

\begin{matrix} (9) & {(d β_{+} / d α)}^{2} + {(d β_{-} / d α)}^{2} = 1 \end{matrix}

The

β_{\pm}

are related to the Kasner exponents

p_{i}

or the

u

parameter of equations (30.5) by

\begin{aligned} d β_{+} / d α & = \frac{1}{2} (1 - 3 p_{3}) \\ = - 1 + (3 / 2) {(1 + u + u^{2})}^{- 1} \\ (10) & d β_{-} / d α & = \frac{1}{2} \sqrt{3} (p_{1} - p_{2}) \\ = - \frac{1}{2} \sqrt{3} (1 + 2 u) {(1 + u + u^{2})}^{- 1} \end{aligned}

Introducing Space Curvature

The first step in generalizing the Kasner metric has focused attention on the "velocity"

β^{'} \equiv (d β_{+} / d α

d β_{-} / d α

) which is a derivative of anisotropy with respect to expansion. The effects of matter or, as will soon appear, space curvature can change the magnitude

‖ β^{'} ‖

from the Kasner value of unity. The second step of generalization is to introduce space curvature. This one achieves in a simple example by retaining the metric components of equation (1), but employing them in a non-holonomic basis. Use the basis vectors introduced in exercises 9.13 and 9.14 on the rotation group

S O (3)

, whose dual 1-forms are

\begin{aligned} σ^{1} = \cos ψ d θ + \sin ψ \sin θ d ϕ, \\ (11) & σ^{2} = \sin ψ d θ - \cos ψ \sin θ d ϕ, \\ σ^{3} = d ψ + \cos θ d ϕ \end{aligned}

to form the metric

\begin{matrix} (12) & d s^{2} = - N^{2} d t^{2} + e^{2 α} {(e^{2 β})}_{i j} σ^{i} σ^{j} \end{matrix}

where

N, α

, and

β_{i j}

are functions of

t

only. When

Box 30.1 (continued)

α = 0 = β_{i j}

, the three-dimensional space metric here reduces to the one studied in exercise 13.15, which is the metric of highest symmetry on the group space

S O (3)

. The simply connected covering space has the 3 -sphere topology, and is obtained by extending the range of the Euler angle

ψ

to give it a

4 π

period

[S U (2)

or spin

\frac{1}{2}

covering of the rotation group]. With

N = 1, \frac{1}{2} a = e^{α}

, and

β_{i j} = 0

, one obtains from equation (12) the same metric (in different coordinates) as that treated in exercise 14.4 and in Chapter 27 in discussions of the closed Friedmann cosmological model. A non-zero value for

β_{i j}

allows the 3 -sphere to have a different circumference on great circles in each of 3 mutually orthogonal principal directions, thus destroying its isotropy but not its homogeneity.

Let us consider only the case with

β_{i j}

diagonal, as in equation (8). Then the

T^{00}

Einstein equation becomes (with

N = 1

as a time-coordinate condition)

\begin{matrix} (13) & 3 ({\dot{α}}^{2} - {\dot{β}}_{+}^{2} - {\dot{β}}_{-}^{2}) + \frac{1}{2} (^{3} R_{IX}) = 8 π T^{00} \end{matrix}

where only the term

\begin{matrix} (14) & ^{3} R_{IX} = \frac{1}{2} e^{- 2 α} trace (2 e^{- 2 β} - e^{4 β}) \end{matrix}

is different from equation (5). This term [see equation (21.92)] is the scalar curvature of a threedimensional slice,

t =

const [which has symmetry properties known as "Bianchi Type IX" for the metric of equations (11) and (12)]. If equation (13) is interpreted in terms of an anisotropy energy density contributing, with

T^{00}

, to the volume expansion

{\dot{α}}^{2}

, then there are not only kinetic energy terms

{\dot{β}}^{2}

[as in equations (5) and (7)], but also a potential energy term. This term shows that negative scalar curvature, which can be produced by anisotropy (

β \neq 0

), is equivalent to a positive potential (or "internal") energy, and suggests that empty space has properties with analogies to an elastic solid and resists shear strains. The more detailed analysis which follows shows that, near
the singularity, the scalar curvature is always negligible when positive.

Negative curvatures, however, arise in this closed universe from large shear

(β)

deformations near the singularity and become large enough to reverse one Kasner shear motion [

u

-value, etc.; equation (10)] and change it to another.

These conclusions and further details of the time-evolution of the "Mixmaster" metric

(11, 12)

require, in principle, the study of all the Einstein equations, not just equation (13) for

T^{00}

. As described in Chapter 21, however, this

T^{00}

constraint equation is central, and actually contains implicitly the full content of the Einstein equations when formulated properly.

Variational Principles

One adequate formulation, adopted here, involves treating equation (13) not as an energy equation (involving velocities), but as a Hamiltonian (involving momenta). Take the Einstein variational principle (21.15) in ADM form (21.95) and carry out the space integration, using

\int σ^{1} \land σ^{2} \land σ^{3} = \int \sin θ d ϕ \land d θ \land d ψ = (4 π)^{2}

, to obtain the action integral in the form

\begin{aligned} (15) & I = & (π) \int {π^{i j} d g_{i j} + N e^{3 α} [^{3} R_{IX} \\ + e^{- 6 α} (\frac{1}{2} {(π^{t})}^{2} - π^{i k} π_{i k})] d t} . \end{aligned}

When introducing the specific form (1) and (8) for

g_{i j}

, it is convenient also to parameterize the diagonal matrix

π^{i}_{k}

as follows:

\begin{aligned} p_{α} & = (2 π) π^{k}_{k} \\ (16) & {p^{i}}_{k} & = (2 π) (π_{k}^{i} - \frac{1}{3} δ_{k}^{i} π_{ℓ}_{ℓ}) \end{aligned}

with

\begin{aligned} 6 p^{1}_{1} = p_{+} + p_{-} \sqrt{3} \\ (17) & 6 p^{2}_{2} = p_{+} - p_{-} \sqrt{3} \\ 6 p_{3}^{3}_{3} = - 2 p_{+} \end{aligned}

[see equation (8)]. The result is

\begin{aligned} I = & \int p_{+} d β_{+} + p_{-} d β_{-} + p_{α} d α \\ - \frac{N e^{- 3 α}}{24 π} [- p_{α}^{2} + p_{+}^{2} + p_{-}^{2} \\ - 24 π^{2} e^{6 α} (^{3} R_{IX})] d t . \end{aligned}

This is cleaned up for further study as follows. Write

\begin{matrix} (18) & ^{3} R_{IX} = \frac{3}{2} e^{- 2 α} (1 - V), \end{matrix}

where

\begin{matrix} (19) & V = V (β) = \frac{1}{3} trace (1 - 2 e^{- 2 β} + e^{4 β}) \end{matrix}

V (0) = 0

; and adjust the zero of

α (α ⟶

α - α_{0}

) so that

e^{2 α} ⟶ (6 π)^{- 1} e^{2 α}

. Then the metric is

\begin{matrix} (20) & d s^{2} = - N^{2} d t^{2} + (6 π)^{- 1} e^{2 α} {(e^{2 β})}_{i j} σ^{i} σ^{j} \end{matrix}

and the variational integral is

\begin{aligned} (21) & I = & \int p_{+} d β_{+} + p_{-} d β_{-} + p_{α} d α \\ - (3 π / 2)^{1 / 2} N e^{- 3 α} H d t, \end{aligned}

with

\begin{matrix} (22) & 2 K \equiv - p_{α}^{2} + p_{+}^{2} + p_{-}^{2} + e^{4 α} (V - 1) \end{matrix}

One demands

δ I = 0

for arbitrary independent variations of

p_{\pm}, p_{α}, β_{\pm}, α, N

to obtain the Einstein equations. From varying

N

, one obtains the fundamental constraint equation

K = 0

[which would reduce to the vacuum version of equation (13) when the momenta are replaced by velocities (via equations obtained by varying the

p

's) if the coordinate condition

N = 1

were imposed.]

ADM Hamiltonian

The standard ADM prescription for reducing this variational principle to canonical (Hamiltonian) form is to choose one of the field variables or momenta as a time-coordinate, and solve the con-
straint for its conjugate Hamiltonian. Here an obvious and satisfactory choice is to set

t = α

, and solve

K = 0

for

H_{ADM} = - p_{α} = {[p_{+}^{2} + p_{-}^{2} + e^{4 α} (V - 1)]}^{1 / 2}

The

\dot{α}

equation [vary

p_{α}

in equation (21)] is

\begin{matrix} (24) & \dot{α} = - (3 π / 2)^{1 / 2} N e^{- 3 α} p_{α} \end{matrix}

and shows that the choice

α = t

(so

\dot{α} = 1

) requires

\begin{matrix} (25) & N_{ADM} = (2 / 3 π)^{1 / 2} e^{3 α} / H_{ADM} . \end{matrix}

The reduced, canonical, variational principle which results when equation (23) is used to eliminate

p_{α}

reads

δ I_{red} = 0

with

\begin{matrix} (26) & I_{red} = \int p_{+} d β_{+} + p_{-} d β_{-} - H_{ADM} d α \end{matrix}

and must be supplemented by equation (25).

Super-Hamiltonian

A more convenient approach here is one more closely related to the Dirac Hamiltonian methods than those of ADM. Note, however, that one does not remove the arbitrariness in the lapse function by taking it to be some specified function

N (t)

of the coordinates. Instead the procedure adopted here is to eliminate

N

from the variational principle (21) by choosing it (coordinate condition!) to be some chosen function of the field variables and momenta,

N = N (α, β_{\pm}, p_{α}, p_{\pm})

. Any such choice, inserted in equation (21), leaves a variational integral in canonical Hamiltonian form. The content of this new variational principle becomes equivalent to the original one only when supplemented by the constraint

\begin{matrix} (27) & K = 0, \end{matrix}

which can no longer be derived from the variational principle. [The other Euler-Lagrange equations for these two principles differ only by terms proportional to

K

, and thus are equivalent when

Box 30.1 (continued)

H = 0

is imposed on the initial conditions.] The choice

\begin{matrix} (28) & N = (2 / 3 π)^{1 / 2} e^{3 α} \end{matrix}

is obvious and convenient. It makes

H

become a super-Hamiltonian in the resulting variational principle

\begin{matrix} (29) & I = \int p_{+} d β_{+} + p_{-} d β_{-} + p_{α} d α - H d λ, \end{matrix}

where

t \equiv λ

has been written to label the specific time-coordinate choice that equation (28) implies.

Mixmaster Dynamics

If matter terms with no additional degrees of freedom are included, the super-Hamiltonian in equation (29) is modified simply. For an example, choose

\begin{matrix} (30) & T^{\hat{0} \hat{0}} = - T_{0}^{0} = (3 / 4)^{2} (μ e^{- 3 α} + Γ e^{- 4 α}) \end{matrix}

for the energy density of matter in a frame with time-axis

e_{\hat{0}} = N^{- 1} (\partial / \partial t)

. The two terms represent a nonrelativistic perfect fluid (

ρ \propto V^{- 1}

) and a radiation fluid (

ρ \propto V^{- 4 / 3}

), respectively, and lead to

\begin{aligned} (31) & 2 K = & - p_{α}^{2} + p_{+}^{2} + p_{-}^{2} + e^{4 α} (V - 1) \\ + μ e^{3 α} + Γ e^{2 α} . \end{aligned}

This Hamiltonian, with its simple quadratic momentum dependence, differs in only two ways from the Hamiltonians of elementary mechanics, namely, (1) in the sign of the

p_{α}^{2}

term and (2) in the detailed shape of the "potential" term as function of

α

and

β_{\pm}

, the study of which reduces to a study of the function

V (β)

. Hamilton's equations, from varying

α, β_{\pm}, p_{α}

, and

p_{\pm}

in equation (29), yield

\begin{matrix} (32) & \frac{d^{2} β_{\pm}}{d λ^{2}} = - \frac{\partial K}{\partial β_{\pm}} = - \frac{1}{2} e^{4 α} \frac{\partial V}{\partial β_{\pm}} \end{matrix}

and

\frac{d^{2} α}{d λ^{2}} = + \frac{\partial K}{\partial α} = 2 e^{4 α} (V - 1) + \frac{3}{2} μ e^{3 α} + Γ e^{2 α}

Thus the sign of the

p_{α}^{2}

term causes

α

to accelerate toward (rather than away from) higher values of the "potential" terms

e^{4 α} (V - 1) + μ e^{3 α} + Γ e^{2 α}

. When

| V | ≪ 1

(small anisotropy), equation (33) is identical to its form in the isotropic Friedmann model, and allows a deceleration only when

α

is large enough that the positive curvature term (

- e^{4 α}

) dominates over matter (

μ e^{3 α}

) and radiation

(Γ e^{2 α})

. Near the singularity

(α ⟶ - \infty)

, the positive curvature term is always negligible compared to radiation and matter.

For studies of the singularity behavior, it is sufficient to study the simplified super-Hamiltonian

\begin{matrix} (34) & 2 K \sim - p_{α}^{2} + p_{+}^{2} + p_{-}^{2} + e^{4 α} V (β), \end{matrix}

since the other terms obviously vanish for

α ⟶

- \infty

. This form retains only the

V

term in

^{3} R_{IX} =

\frac{3}{2} e^{- 2 α} (1 - V)

, which dominates when the curvature of this closed universe becomes negative,

V ≫ 1

. If the term in

V (β)

were also negligible, then

K = - p_{α}^{2} + p_{+}^{2} + p_{-}^{2}

would make each

p_{α}, p_{\pm}

constant, giving the Kasner behavior with

d β_{\pm} / d α = p_{\pm} / p_{α} = const

and

| d β / d α |^{2} = 1

as expected (since matter and curvature have been neglected). To proceed further, a study of

V (β)

is required, based on equations (19) and (8), and their immediate consequence:

\begin{aligned} V (β) = & \frac{1}{3} e^{- 8 β_{+}} - \frac{4}{3} e^{- 2 β_{+}} \cosh 2 \sqrt{3} β_{-} \\ (35) & + 1 + \frac{2}{3} e^{4 β_{+} (\cosh 4 \sqrt{3} β_{-} - 1)} \end{aligned}

One finds that

V (β)

is a positive definite "potential well" which has the same symmetries as an equi-
lateral triangle in the

β_{+} β_{-}

plane. Near the origin,

β_{\pm} = 0

, the equipotentials are circles, since

\begin{matrix} (36) & V (β) = 8 (β_{+}^{2} + β_{-}^{2}) + 0 (β^{3}) \end{matrix}

For large

β

values, one finds

\begin{matrix} (37) & V (β) \sim \frac{1}{3} e^{- 8 β_{+}}, β_{+} ⟶ - \infty \end{matrix}

and

\begin{matrix} (38) & V (β) \sim 1 + 16 β_{-}^{2} e^{4 β_{+}}, \begin{array}{ll} β_{+} ⟶ + \infty \\ | β_{-} | ≪ 1 \end{array} \end{matrix}

These two asymptotic forms, together with the triangular symmetry, give a complete asymptotic description of

V (β)

, as sketched in the figure, where on successive levels separated by

Δ β = 1

, the potential

V

increases by a factor of

e^{8} =

3 \times 10^{3}

"Bounce" Interrupts Kasner-like Steps Toward the Singularity

The dominant feature of the

V (β)

potential is evidently its steep (exponential) triangular walls, with equation (37) representing the typical one for study. Under the influence of this potential wall, the evolution of this model universe is governed by the super-Hamiltonian

\begin{matrix} (39) & 2 H \sim - p_{α}^{2} + p_{+}^{2} + p_{-}^{2} + \frac{1}{3} e^{4 (α - 2 β_{+})} \end{matrix}

α ⟶ - \infty

with

d β_{+} / d α > 1 / 2

[recall

d β_{+} / d α

=

const.,

| d β / d α | = 1

, when the last term in (39) is small], then the potential term grows and will eventually become large enough to influence the motion. A simple "Lorentz" transformation, suggested by the superspace metric (coefficients of the

Some equipotentials, $V (β) =$ constant, are shown for the function defined in equation (35). Equipotentials near the origin of the $β$ -plane are closed curves for $V < 1$ and are omitted here.

Box 30.1 (continued)

quadratic in the momenta) simplifies the computation further. Set

\begin{aligned} {\bar{β}}_{+} & = (β_{+} - \frac{1}{2} α) / \sqrt{3 / 4} \\ \bar{α} & = (α - \frac{1}{2} β_{+}) / \sqrt{3 / 4} \end{aligned}

and find

\begin{matrix} (40) & 2 K = - {\bar{p}}_{α}^{2} + {\bar{p}}_{+}^{2} + p_{-}^{2} + \frac{1}{3} \exp (- 4 \sqrt{3} {\bar{β}}_{+}) . \end{matrix}

For this super-Hamiltonian both

{\bar{p}}_{α}

and

p_{-}

are constants of motion, whereas the

{\bar{β}}_{+}

-Hamiltonian,

{\bar{p}}_{+}^{2} + \frac{1}{3} e^{- 4 \sqrt{3} {\bar{β}}_{+}}

, represents a simple bounce against a one-dimensional potential wall with the initial and final values of

{\bar{p}}_{+}

different only in sign. The behavior of the anisotropy parameters

β_{\pm}

near the singularity thus consists of a simple Kasner step (where

d β_{\pm} / d α =

const., with the

d β_{+} / d α \geq \frac{1}{2}

, or conditions equivalent by symmetry, satisfied relative to one of the three walls), followed by a bounce against that wall, beginning a new Kasner step with other Kasner parameters. [The most detailed description of this behavior and its relation to more general cosmological models can be found in Belinsky, Khalatnikov, and Lifshitz (1970)-see also the briefer report, Khalatnikov and Lifshitz (1970)-using quite different methods. For detailed developments by Hamiltonian methods, which supercede the partial Lagrangian methods of Misner (1969b), see Misner (1970, 1972a), and Ryan (1972a,b).]

Steady-State, Quasiperiodic Infinity of "Bounces" Approaching the Singularity

Some comprehensive features of the singularity behavior, involving many Kasner-like steps, can be exhibited by another transformation of the parameter space (superspace) of the metric field.

The transformation introduces a "radial"

t

coordinate out from the origin of

α β_{\pm}

space, while respecting the metric properties of this superspace implied by the form of the super-Hamiltonian. Thus one defines (for any constant

α_{0}

)

\begin{aligned} α_{0} - α & = e^{t} \cosh ζ, \\ (41) & β_{+} & = e^{t} \sinh ζ \cos ϕ, \\ β_{-} & = e^{t} \sinh ζ \sin ϕ, \end{aligned}

and finds

2 K = e^{- 2 t} [(- p_{t}^{2} + p_{ξ}^{2} + p_{ϕ}^{2} \sinh^{- 2 ζ}) + e^{2 t} e^{4 α} V]

The advantage of this transformation is that in the limit

t ⟶ \infty (α ⟶ - \infty

, singularity) the potential terms become, in first approximation, independent of

t

. Thus equation (37) gives, for one potential wall,

\begin{aligned} e^{2 t} e^{4 α} V \sim & \frac{1}{3} e^{2 t} \exp [4 α_{0} \\ (43) & - 8 e^{t} (\sinh ζ \cos ϕ + \frac{1}{2} \cosh ζ)] . \end{aligned}

For

t ⟶ \infty

this expression evidently tends to either zero or infinity, depending on the sign of the expression in parentheses. Therefore define the asymptotic potential walls by

\begin{matrix} (44) & \tanh ζ + \frac{1}{2} \sec ϕ = 0 \end{matrix}

in the sector

| ϕ - π | < π / 3

, and equivalent formulae in which

ϕ

is replaced by

ϕ \pm (2 π / 3)

for the other sides of the triangle. Consequently, an asymptotic approximation to the super-Hamiltonian is

\begin{matrix} (45) & 2 K = e^{- 2 t} [- p_{t}^{2} + p_{ξ}^{2} + p_{ϕ}^{2} \sinh^{- 2 ζ} + V^{'} (ζ, ϕ)], \end{matrix}

where

V^{'} (ζ, ϕ)

vanishes inside the asymptotic walls (44) and equals

+ \infty

outside. Because the remaining

t

-dependence is a common factor in (45), a simple change of independent variable

e^{- 2 t} d λ =

d λ^{'}

in equation (29)-equivalent to the choice

\begin{matrix} (46) & N = (2 / 3 π)^{1 / 2} e^{- 2 t} \exp [3 (α_{0} - e^{t} \cosh ζ)] \end{matrix}

in place of equation (28)-gives a new superHamiltonian

K^{'} = e^{2 t} K

with the variational integral

\begin{matrix} (47) & I = \int p_{t} d t + p_{s} d ξ + p_{ϕ} d ϕ - K^{'} d λ^{'} . \end{matrix}

In the asymptotic approximation where

\begin{matrix} (48) & 2 K^{'} = - p_{t}^{2} + p_{ξ}^{2} + p_{ϕ}^{2} \sinh^{- 2 ζ} + V^{'} (ζ, ϕ), \end{matrix}

one immediately sees that

p_{t}

is a constant of motion, and that the "bouncing" of the

ζ ϕ

values within the asymptotic potential walls is a stationary, quasi-periodic process in this time-coordinate

λ^{'}

(or

t

, since

d t / d λ^{'} = - p_{t} =

const). [More detailed studies based on this asymptotic superHamiltonian show that the motion is even ergodic, with

ζ ϕ

approaching arbitrarily close to any given value infinitely many times as

t ⟶ \infty

; see Chitre (1972a).]

Summary

One has found the singularity behavior in this Mixmaster example to be extraordinarily active. In the simple Kasner singularity, two axes collapse, but the third is stretched in a simple tidal deformation accompanied by volume compression. But in the Mixmaster example, every such collapse attempt is defeated by the high negative curvature it implies. Or rather it is diverted to another attempt as compression continues inexorably, but the tidal deformations attempt first one configuration, then another, in an infinitely recurring probing of all possible configurations.

Speculations on Time and the Singularity

The cosmological singularity (in all examples where its character is not known to be unstable) involves infinite curvature and infinite density. One's abhorence of such a theoretical prediction is particularly heightened by the correlative prediction that these infinities occurred at a finite proper time in the past, and would-if they
recur-occur again at some finite proper time in the future. The singularity prediction would be more tolerable if the infinite densities could be removed to the infinitely distant past. The universe could then, as now, find its natural state to be one of expansion, so every finite density will have been experienced at some suitably remote past time, but infinite density becomes a formal abstraction never realized in the course of evolution.

To push infinite curvature out of the finite past might be achieved in two ways. It is not known which, if either, works. One way is to change the physical laws which require the singularity, changing them perhaps only in obvious and desirable ways, such as stating the laws of gravity in a proper quantum language. Computations of quantum geometry are not yet definitive, however, and some (perhaps inadequate) approximations [Misner (1972a)] do not remove the singularity problem.

Another way to discard the singularity is to accept the mathematics of the classical Einstein equations, but reinterpret it in terms of an infinite past time. There are, of course, simple and utterly inadequate ways to do this by arbitrary coordinate transformations such as

t = \ln τ

which change a

τ = 0

singularity into one at

t = - \infty

. But an arbitrary coordinate is without significance. The problem is that the singularity occurs at a finite proper time in the past, and proper time is the most physically significant, most physically real time we know. It corresponds to the ticking of physical clocks and measures the natural rhythms of actual events. To reinterpret finite past time as infinite, one must attack proper time on precisely these grounds, and claim it is inadequately physical. On a local basis, where special relativity is valid, no challenge to the physical significance of proper time can succeed. It is on a more global scale that the physical primacy of proper time needs to be reviewed.
"The cosmological singularity occurred ten thousand million years ago." In this statement, take time to mean the proper time along the world line of the solar system, ephemeris time. Then the statement would have a most direct physical sig-

Box 30.1 (continued)

nificance if it meant that the Earth had completed

10^{10}

orbits about the sun since the beginning of the universe. But proper time is not that closely tied to actual physical phenomena. The statement merely implies that those

5 \times 10^{9}

orbits which the earth may have actually accomplished give a standard of time which is to be extrapolated in prescribed ways, thus giving theoretical meaning to the other

5 \times 10^{9}

years which are asserted to have preceeded the formation of the solar system.

A hardier standard clock changes the details of the argument, but not its qualitative conclusion. To interpret

10^{10}

years in terms of SI (Systeme Internationale) seconds assigns a past history containing some

3 \times 10^{27}

oscillations of a hyperfine transition in neutral Cesium. But again the critical early ticks of the clock (needed to locate the singularity in time by actual physical events) are missing. The time needed for stellar nucleosynthesis to produce the first Cesium disqualifies this clock on historical grounds, and the still earlier high temperatures nearer the singularity would have ionized all Cesium even if this element had predated stars.

Thus proper time near the singularity is not a direct counting of simple and actual physical phenomena, but an elaborate mathematical extrapolation. Each actual clock has its "ticks" discounted by a suitable factor-

3 \times 10^{7}

seconds per orbit from the Earth-sun system,

1.1 \times 10^{- 10}

seconds per oscillation for the Cesium transition, etc. Since no single clock (because of its finite size and strength) is conceivable all the way back to the singularity, a statement about the proper time since the singularity involves the concept of an infinite sequence of successively smaller and sturdier clocks with their ticks then discounted and
added. "Finite proper time," then, need not imply that any finite sequence of events was possible. It may describe a necessarily infinite number of events ("ticks") in any physically conceivable history, converted by mathematics into a finite sum by the action of a non-local convergence factor, the "discount" applied to convert "ticks" into "proper time."

Here one has the conceptual inverse of Zeno's paradox. One rejects Zeno's suggestion that a single swing of a pendulum is infinitely complicatedbeing composed of a half period, plus a quarter period, plus

2^{- n}

ad infinitum-because the terms in his infinite series are mathematical abstractions, not physically achieved discrete acts in a drama that must be played out. By a comparable standard, one should ignore as a mathematical abstraction the finite sum of the proper-time series for the age of the universe, if it can be proved that there must be an infinite number of discrete acts played out during its past history. In both cases, finiteness would be judged by counting the number of discrete ticks on realizable clocks, not by assessing the weight of unrealizable mathematical abstractions.
Whether the universe is infinitely old by this standard remains to be determined. The quantum influences, in particular, remain to be calculated. The decisive question is whether each presentepoch event is subject to the influence of infinitely many previous discrete events. In that case statistical assumptions (large numbers, random phases, etc.) could enter in stronger ways into theories of cosmology. The Mixmaster cosmological model does have an infinite past history in this sense, since each "bounce" from one Kasner-like motion to another is a recognizable cosmological event, of which infinitely many must be realized between any finite epoch and the singularity.

§30.8. HORIZONS AND THE ISOTROPY OF the microwave background

The fundamental cosmological question-"Must a universe that is born chaotic necessarily become as homogeneous and isotropic as our universe is, and do so before life evolves?"-entails one further issue. This issue is horizons. As was discussed in §27.10, at any given epoch in the expansion of a Friedmann universe (e.g., the present epoch), there may be significant portions of the universe from which no light signal or other causally propagating influence will have yet reached Earth in the time available since the initial singularity. "If we should live so long," the question would arise, "will the new portions of the universe which first come into view during the next ten billion years look statistically identical to the neighboring portions which are already being seen?"

Fortunately, this question need not be posed only for the future. It can be asked as of some past time, and the answer then is yes. Microwave background radiation arrives at the earth from all directions in the sky with very nearly the same temperature. [The data of Boughn, Fram, and Partridge (1971) and of Conklin (1969) show

Δ T / T ≦ 0.004

.] The plasma that emitted the microwave radiation coming to earth from one direction in the sky had not been able, before the epoch of emission, to communicate causally with the plasma emitting the radiation that arrives from other directions. If one adopts a Friedmann model of the universe, then different sectors of the microwave sky are disjoint from each other in this sense if they are separated from each other by more than

30^{\circ}

, even if the microwaves were emitted as recently as

z = 7

. (The critical angle is much smaller if the microwaves were last scattered at

z = 1, 000

.) From this, one concludes that the foundations for the homogeneity and isotropy of the universe were laid long before the universe became approximately Friedmann, for if statistical homogeneity and isotropy of the universe had not already been achieved at the longest wavelengths earlier, these horizon limitations would have prevented any further synchronization of conditions over large scales while the universe was in a nearly Friedmann state, and small amplitude (

10 %

) deviations from isotropy should be observed now.

The mixmaster universe received its name from the hope that it could contribute to the solution of this problem. The very large

u

values that occur sporadically an infinite number of times near the singularity in a mixmaster universe give a geometry close to that of the Kasner model with

p_{1} = 1, p_{2} = p_{3} = 0

. This model can be written in the form

\begin{matrix} (30.6) & d s^{2} = e^{2 η} (- d η^{2} + d x^{2}) + d y^{2} + d z^{2} \end{matrix}

where

η = \ln t

. If this metric is converted into a closed-universe model by interpreting

x, y, z

as angle coordinates each with period

4 π

, then one sees that a light ray can circumnavigate the universe in the

x

-direction in a time interval

Δ η = 4 π

, which corresponds to a volume expansion by a factor

\sqrt{- g_{1}} / \sqrt{- g_{2}} = e^{4 π}

. Unfortunately, a quantitative analysis of the degree and frequency with which the mixmaster universe achieves this specific Kasner form suggests that the horizon breaking

Horizons in a Friedmann universe

Observed isotropy of microwave radiation proves foundations for homogeneity were laid before universe became Friedmann-like

What made the universe homogeneous and isotropic?
(1) Mixmaster oscillations? probably not
(2) particle creation near singularity?
is inadequate to explain the present state of the universe [Doroshkevich, Lukash, and Novikov (1971); Chitre (1972)]. It may turn out that particle creation near the singularity can solve this horizon question, as well as provide for the dissipation of anisotropy. Hope is provided by the fact that particle creation, when described in purely classical terms, has some acausal appearances, even though it is a strictly causal process at the quantum level [Zel'dovich (1972)].

GRAVITATIONAL COLLAPSE AND BLACK HOLES

Wherein the reader is transported to the land of black holes, and encounters colonies of static limits, ergospheres, and horizons-behind whose veils are hidden gaping, ferocious singularities.

синотен 31

SCHWARZSCHILD GEOMETRY

§31.1. INEVITABILITY OF COLLAPSE FOR MASSIVE STARS

There is no equilibrium state at the endpoint of thermonuclear evolution for a star containing more than about twice the number of baryons in the sun

(A > A_{max} \sim 2 A_{⊙})

. This is one of the most surprising-and disturbing-consequences of the discussion in Chapter 24. Stated differently: A star with

A > A_{max} \sim 2 A_{⊙}

must eject all but

A_{max}

of its baryons-e.g., by nova or supernova explosions-before settling down into its final resting state; otherwise there will be no final resting state for it to settle down into.

What is the fate of a star that fails to eject its excess baryons before nearing the endpoint of thermonuclear evolution? For example, after a very massive supernova explosion, what will become of the collapsed degenerate-neutron core when it contains more than

A_{max}

baryons? Such a supercritical mass cannot explode, since it is gravitationally bound and it has no more thermonuclear energy to release. Nor can it reach a static equilibrium state, since there exists no such state for so large a mass. There remains only one alternative; the supercritical mass must collapse through its "gravitational radius,"

r = 2 M

, leaving behind a gravitating "black hole" in space.
The phenomenon of collapse through the gravitational radius, as described by classical general relativity, will be the subject of the next chapter. However, before tackling it, one must understand more fully than heretofore the Schwarzschild spacetime geometry, which surrounds black holes and collapsing stars as well as static stars.

This chapter will concern itself with two topics that, at first sight, appear to be disconnected. One is the fall of a test particle in a preexisting Schwarzschild geometry, which is regarded as static, but can also be visualized as all that remains of a star that underwent collapse some time ago. The second topic is the physical

This chapter, on Schwarzschild geometry, is key preparation for understanding gravitational collapse (next chapter) and black holes (following chapter)
character of this geometry, regarded in and by itself. For the exploration of this geometry, the test particle serves as the best of all explorers. But the test particle may also be regarded in another light. It can be viewed as a rag-tag johnny-comelately piece of the matter of the falling star. Regarded in this way, it provides the simplest of all illustrations of an asymmetry in the distribution of mass of a collapsing star. That this asymmetry irons itself out will therefore give one some preliminary insight into how more complicated asymmetries also iron themselves out. In brief, the motion of the test particle and the dynamics of the Schwarzschild geometry (for this geometry will prove to be dynamic), two apparently different problems, have the happy ability to throw light on each other.

§31.2. THE NONSINGULARITY OF THE GRAVITATIONAL RADIUS

The Schwarzschild spacetime geometry

\begin{matrix} (31.1) & d s^{2} = - (1 - \frac{2 M}{r}) d t^{2} + \frac{d r^{2}}{1 - 2 M / r} + r^{2} (d θ^{2} + \sin^{2} θ d ϕ^{2}) \end{matrix}

appears to behave badly near

r = 2 M

; there

g_{t t}

becomes zero, and

g_{r r}

becomes infinite. However, one cannot be sure without careful study whether this pathology in the line element is due to a pathology in the spacetime geometry itself, or merely to a pathology of the

(t, r, θ, ϕ)

coordinate system near

r = 2 M

. (As an example of a coordinate-induced pathology, consider the neighborhood of

θ = 0

on one of the invariant spheres,

t =

const and

r =

const. There

g_{ϕ ϕ}

becomes zero because the coordinate system behaves badly; however, the intrinsic, coordinate-independent geometry of the sphere is well-behaved there. For another example, see Figure 1.4.

The worrisome region of the Schwarzschild geometry,

r = 2 M

, is called the "gravitational radius," or the "Schwarzschild radius," or the "Schwarzschild surface," or the "Schwarzschild horizon," or the "Schwarzschild sphere." It is also called the "Schwarzschild singularity" in some of the older literature; but that is a misnomer, since, as will be shown, the spacetime geometry is not singular there.

To determine whether the spacetime geometry is singular at the gravitational radius, send an explorer in from far away to chart it. For simplicity, let him fall freely and radially into the gravitational radius, carrying his orthonormal tetrad with him as he falls. His trajectory through spacetime ["parabolic orbit"; radial geodesic of metric (31.1)] is

\begin{matrix} \frac{τ}{2 M} = - \frac{2}{3} {(\frac{r}{2 M})}^{3 / 2} + constant \\ (31.2) & \frac{t}{2 M} = - \frac{2}{3} {(\frac{r}{2 M})}^{3 / 2} - 2 {(\frac{r}{2 M})}^{1 / 2} + \ln | \frac{(r / 2 M)^{1 / 2} + 1}{(r / 2 M)^{1 / 2} - 1} | + constant. \end{matrix}

[See

$ 25.5

and especially equation (25.38) for derivation and discussion.] One obtains the

r

coordinate of the explorer in terms of the proper time measured on a clock
he carries,

r (τ)

, by inverting the first equation; one finds his

r

coordinate in terms of coordinate time,

r (t)

, by inverting the second equation.

Of all the features of the traveler's trajectory, one stands out most clearly and disturbingly: to reach the gravitational radius,

r = 2 M

, requires a finite lapse of proper time, but an infinite lapse of coordinate time:

\begin{matrix} r / 2 M = 1 - (τ + constant) / 2 M when near r = 2 M \\ (31.3) & r / 2 M = 1 + constant \times \exp (- t / 2 M) in limit as t ⟶ \infty \end{matrix}

(see Fig. 25.5.) Of course, proper time is the relevant quantity for the explorer's heart-beat and health. No coordinate system has the power to prevent him from reaching

r = 2 M

. Only the coordinate-independent geometry of spacetime could possibly do that; and equation (31.3) shows it does not!

Let the explorer approach and reach

r = 2 M

, then. What spacetime geometry does he measure there? Is it singular or nonsingular? Restated in terms of measurements, do infinite tidal gravitational forces tear the traveler apart and crush him as he approaches

r = 2 M

, or does he feel only finite tidal forces which in principle his body can withstand?

The tidal forces felt by the explorer as he passes a given radius

r

are measured by the components of the Riemann curvature tensor with respect to his orthonormal frame there (equation of geodesic deviation). To calculate those curvature components at

r

, proceed in two steps. (1) Calculate the components, not in the traveler's frame, but rather in the "static" orthonormal frame

\begin{matrix} (31.4a) & ω^{\hat{t}} = {(1 - \frac{2 M}{r})}^{1 / 2} d t, ω^{\hat{ı}} = \frac{d r}{(1 - 2 M / r)^{1 / 2}}, ω^{\hat{θ}} = r d θ, ω^{\hat{ϕ}} = r \sin θ d ϕ \end{matrix}

located at the event through which he is passing; the result [obtainable from equations (14.50) and (14.51) by setting

e^{2 ϕ} = e^{- 2 A} = 1 - 2 M / r]

\begin{matrix} (31.4b) & \begin{array}{cl} R_{\hat{t} \hat{t} \hat{r}} = \frac{- 2 M}{r^{3}}, & R_{\hat{θ} \hat{θ} \hat{t} \hat{θ}} = R_{\hat{t} \hat{ϕ} \hat{t} \hat{ϕ}} = \frac{M}{r^{3}} \\ R_{\hat{θ} \hat{ϕ} \hat{θ} \hat{ϕ}} = \frac{2 M}{r^{3}}, & R_{\hat{r} \hat{θ} \hat{r} \hat{θ}} = R_{\hat{r} \hat{ϕ} \hat{r} \hat{ϕ}} = \frac{- M}{r^{3}} \end{array} \end{matrix}

all other

R_{\hat{α} \hat{β} \hat{γ} \hat{δ}}

vanish except those obtainable from the above by symmetries of Riemann.
(2) Calculate the components in the explorer's frame by applying to the "staticframe" components (31.4b) the appropriate transformation-for

r > 2 M

, a Lorentz boost in the

e_{\hat{r}}

direction with ordinary velocity

v^{\hat{r}}

; for

r < 2 M

, not a "boost," but a transformation given by the standard boost formula (Box 2.4) with

v^{\hat{r}} > 1

. Here

\begin{matrix} (31.5) & v^{\hat{r}} = \frac{{(g_{r r})}^{1 / 2} d r}{{(- g_{t t})}^{1 / 2} d t} = \frac{d r / d t}{1 - 2 M / r} = - {(\frac{2 M}{r})}^{1 / 2} \end{matrix}

The amazing result (a consequence of special algebraic properties of the Schwarzschild geometry, and somewhat analogous to what happens-or, rather, does not hap-

An infalling observer reaches

r = 2 M

in finite proper time but infinite coordinate time
pen-to the components of the electromagnetic field,

E

and

B

, when they are both parallel to a boost) is this: all the components of Riemann are left completely unaffected by the boost. If

e_{\hat{ρ}}

is the traveler's radial basis vector, and

e_{\hat{τ}} = u

is his time basis vector, then

\begin{aligned} (31.6) & R_{\hat{γ} \hat{ρ} \hat{τ} \hat{ρ}} = - 2 M / r^{3}, & R_{\hat{γ} \hat{θ} \hat{τ} \hat{θ}} = R_{\hat{τ} \hat{ϕ} \hat{ϕ} \hat{ϕ}} = M / r^{3}, \\ R_{\hat{θ} \hat{ϕ} \hat{θ} \hat{ϕ}} = 2 M / r^{3}, & R_{\hat{ρ} \hat{ϕ} \hat{ρ} \hat{θ}} = R_{\hat{ρ} \hat{ϕ} \hat{ρ} \hat{ϕ} \hat{}} = - M / r^{3} . \end{aligned}

(See exercise 31.1.)
The payoff of this calculation: according to equations (31.6), none of the components of Riemann in the explorer's orthonormal frame become infinite at the gravitational radius. The tidal forces the traveler feels as he approaches

r = 2 M

are finite; they do not tear him apart-at least not when the mass

M

is sufficiently great, because at

r = 2 M

the typical non-zero component

R_{\hat{α} \hat{β} \hat{γ} δ}

of the curvature tensor is of the order

1 / M^{2}

. The gravitational radius is a perfectly well-behaved, nonsingular region of spacetime, and nothing there can prevent the explorer from falling on inward.

By contrast, deep inside the gravitational radius, at

r = 0

, the traveler must encounter infinite tidal forces, independently of the route he uses to reach there. One says that "

r = 0

is a physical singularity of spacetime." To see this, one need only calculate from equation (31.4b) or (31.6) the "curvature invariant":

\begin{matrix} (31.7) & I \equiv R_{\hat{α} \hat{β} \hat{γ} \hat{δ}} R^{\hat{α} \hat{β} \hat{γ} \hat{δ}} = 48 M^{2} / r^{6} \end{matrix}

Box 31.1 THE "SCHWARZSCHILD SINGULARITY": HISTORICAL REMARKS

Although Eddington (1924) was the first to construct a coordinate system that is nonsingular at

r = 2 M

, he seems not to have recognized the significance of his result. Lemaître (1933c, especially p. 82) appears to have been the first to recognize that the so-called "Schwarzschild singularity" at

r = 2 M

is not a singularity. He wrote, "La singularité du champ de Schwarzschild est donc une singularité fictive, analogue à celle qui se présentait a l'horizon du centre dans la forme originale de l'univers de de Sitter". He also provided a coordinate system to go through

r = 2 M

. However, his coordinate system, like Eddington's, covered only half of the Schwarzschild geometry:
regions I and II of Figure 31.3. Synge (1950) was the first to discover the incompleteness in the Eddington and Lemaittre coordinate systems, and to provide coordinates that cover the entire geometry (regions I, II, III, IV of Figure 31.3). Fronsdal (1959), unaware of Synge's work, rediscovered the global structure of the Schwarzschild geometry by means of embedding diagrams and calculations. The coordinate system that provides maximum insight into the Schwarzschild geometry is the one generally known as the Kruskal-Szekeres coordinate system. It was constructed independently by Kruskal (1960) and by Szekeres (1960).

In every local Lorentz frame this will be a sum of products of curvature components, and it will have the same value

48 M^{2} / r^{6}

. Thus, in every local Lorentz frame, including the traveler's, Riemann will have one or more infinite components as

r ⟶ 0

; i.e., tidal forces will become infinite.

Exercise 31.1. TIDAL FORCES ON INFALLING EXPLORER

(a) Carry out the details of the derivation of the Riemann tensor components (31.6).
(b) Calculate, roughly, the critical mass

M_{crit}

such that, if

M > M_{crit}

the explorer's body (a human body made of normal flesh and bones) can withstand the tidal forces at

r = 2 M

, but if

M < M_{crit}

his body is mutilated by them. [Answer:

M_{crit} \sim 1000 M_{⊙}

. Evidently, if

M \sim M_{⊙}

the physicist should transform himself into an ant before taking the plunge! For details see

§ 32.6

r = 0

the curvature is infinite

EXERCISE

§31.3. BEHAVIOR OF SCHWARZSCHILD COORDINATES AT $r = 2 M$

Since the spacetime geometry is well behaved at the gravitational radius, the singular behavior there of the Schwarzschild metric components,

g_{t t} = - (1 - 2 M / r)

and

g_{r r} = (1 - 2 M / r)^{- 1}

, must be due to a pathology there of the Schwarzschild coordinates

t, r, θ, ϕ

. Somehow one must find a way to get rid of that pathology-i.e., one must construct a new coordinate system from which the pathology is absent. Before doing this, it is helpful to understand better the precise nature of the pathology.
The most obvious pathology at

r = 2 M

is the reversal there of the roles of

t

and

r

as timelike and spacelike coordinates. In the region

r > 2 M

, the

t

direction,

\partial / \partial t

, is timelike

(g_{t t} < 0)

and the

r

direction,

\partial / \partial r

, is spacelike

(g_{r r} > 0)

; but in the region

r < 2 M, \partial / \partial t

is spacelike

(g_{t t} > 0)

and

\partial / \partial r

is timelike

(g_{r r} < 0)

What does it mean for

r

to "change in character from a spacelike coordinate to a timelike one"? The explorer in his jet-powered spaceship prior to arrival at

r = 2 M

always has the option to turn on his jets and change his motion from decreasing

r

(infall) to increasing

r

(escape). Quite the contrary is the situation when he has once allowed himself to fall inside

r = 2 M

. Then the further decrease of

r

represents the passage of time. No command that the traveler can give to his jet engine will turn back time. That unseen power of the world which drags everyone forward willy-nilly from age twenty to forty and from forty to eighty also drags the rocket in from time coordinate

r = 2 M

to the later value of the time coordinate

r = 0

. No human act of will, no engine, no rocket, no force (see exercise 31.3) can make time stand still. As surely as cells die, as surely as the traveler's watch ticks away "the unforgiving minutes," with equal certainty, and with never one halt along the way,

r

drops from

2 M

to 0 .

r = 2 M

, where

r

and

t

exchange roles as space and time coordinates,

g_{t t}

vanishes while

g_{r r}

is infinite. The vanishing of

g_{t t}

suggests that the surface

r = 2 M

, which

Nature of the coordinate pathology at

r = 2 M

:
(1)

t

and

r

reverse roles as timelike and spacelike coordinates
(2) the region

r = 2 M

- \infty < t < + \infty

is two-dimensional rather than three
(3) radial geodesics reveal that the regions

r = 2 M

t = \pm \infty

are "finite" parts of spacetime
appears to be three-dimensional in the Schwarzschild coordinate system

(- \infty < t <

+ \infty, 0 < θ < π, 0 < ϕ < 2 π

) has zero volume and thus is actually only two-dimensional, or else is null; thus,

\begin{aligned} (31.8) & \int_{r = 2 M} {| g_{t t} g_{θ θ} g_{ϕ ϕ} |}^{1 / 2} d t d θ d ϕ = 0 \\ \int_{(r = 2 M, t = const)} {| g_{θ θ} g_{ϕ ϕ} |}^{1 / 2} d θ d ϕ = 4 π (2 M)^{2} \end{aligned}

The divergence of

g_{r r}

r = 2 M

does not mean that

r = 2 M

is infinitely far from all other regions of spacetime. On the contrary, the proper distance from

r = 2 M

to a point with arbitrary

r

\begin{matrix} (31.9) & \int_{2 M}^{r} {| g_{r r} |}^{1 / 2} d r = {\begin{array}{r} [r (r - 2 M)]^{1 / 2} + 2 M \ln | (r / 2 M - 1)^{1 / 2} + (r / 2 M)^{1 / 2} | \\ when r > 2 M \\ - 2 M \cot^{- 1} [r^{1 / 2} / (2 M - r)^{1 / 2}] - [r (2 M - r)]^{1 / 2} \\ when r < 2 M \end{array} \end{matrix}

which is finite for all

0 < r < \infty

.
Just how the region

r < 2 M

is physically connected to the region

r > 2 M

can be discovered by examining the radial geodesics of the Schwarzschild metric. Focus attention, for concreteness, on the trajectory of a test particle that gets ejected from the singularity at

r = 0

, flies radially outward through

r = 2 M

, reaches a maximum radius

r_{max}

("top of orbit") at proper time

τ = 0

and coordinate time

t = 0

, and then falls back down through

r = 2 M

r = 0

. The solution of the geodesic equation for such an orbit was derived in

§ 25.5

and described in Figure 25.3. It has the "cycloid form" (with the parameter

η

running from

- π

+ π

\begin{matrix} (31.10a) & r = \frac{1}{2} r_{max} (1 + \cos η) \\ (31.10b) & τ = {(r_{max}^{3} / 8 M)}^{1 / 2} (η + \sin η) \\ (31.10c) & t = 2 M \ln | \frac{{(r_{max} / 2 M - 1)}^{1 / 2} + \tan (η / 2)}{{(r_{max} / 2 M - 1)}^{1 / 2} - \tan (η / 2)} | \\ + 2 M {(\frac{r_{max}}{2 M} - 1)}^{1 / 2} [η + (\frac{r_{max}}{4 M}) (η + \sin η)] \end{matrix}

Figure 31.1 plots this orbit in the

r, t

-coordinate plane (curve

F - F^{'} - F^{''}

), along with several other types of radial geodesics.

Every radial geodesic except a "set of geodesics of measure zero" crosses the gravitational radius at

t = + \infty

(or at

t = - \infty

, or both), according to Figure 31.1 and the calculations behind that figure (exercises for the student! See Chapter 25). One therefore suspects that all the physics at

r = 2 M

is consigned to

t = \pm \infty

by reason of some unhappiness in the choice of the Schwarzschild coordinates. A better coordinate system, one begins to believe, will take these two "points at infinity" and

Figure 31.1.
Typical radial geodesics of the Schwarzschild geometry, as charted in Schwarzschild coordinates (schematic).

F F^{'} F^{''}

[see equations (31.10)] is the timelike geodesic of a test particle that starts at rest at

r = 5.2 M

and falls straight in, arriving in a finite proper time at the singularity

r = 0

(zig-zag marking). The unhappiness of the Schwarzschild coordinate system shows in two ways: (1) in the fact that

t

goes to

\infty

partway through the motion; and (2) in the fact that

t

thereafter decreases as

τ

(not shown) continues to increase. The course of the same trajectory prior to

t = 0

may be constructed by reflecting the diagram in the horizontal axis ("time inversion"). The time-reversed image of

F

" marks the ejection of the test particle from the singularity.

A A^{'} A^{''}

is a timelike geodesic which comes in from

r = + \infty \cdot B B^{'} B^{''}

is the null geodesic travelled by a photon that falls straight in (no summit; never at rest!).

D D^{'} D^{''}

is a spacelike radial geodesic. So is

C C^{'}

, but

E^{'} E^{''}

is timelike. Neither of the latter two ever succeed in crossing

r = 2 M

. (Unanswered questions about these geodesics will answer themselves in Figure 31.4, where the same world lines are charted in a "Kruskal-Szekeres diagram").

Described mathematically via equation (31.10), the geodesic

F_{inverse}^{''} F_{inverse}^{'} F F^{'} F^{''}

starts with ejection at

r = 0 at t = - 2 π M {(\frac{r_{max}}{2 M} - 1)}^{1 / 2} (\frac{r_{max}}{4 M} + 1), τ = - \frac{π}{2} {(\frac{r_{max}^{3}}{2 M})}^{1 / 2}

it flies outward with increasing proper time

τ

, but decreasing coordinate time,

t

, until it reaches the gravitational radius

r = 2 M at t = - \infty, τ = - {(\frac{r_{max}^{3}}{8 M})}^{1 / 2} \cos^{- 1} (\frac{4 M}{r_{max}} - 1) - r_{max} {(1 - \frac{2 M}{r_{max}})}^{1 / 2}

it then continues to fly on outward, but with coordinate time now increasing from

t = - \infty

, until it reaches its maximum radius

r = r_{max} at t = 0, τ = 0 (event F in diagram)

it then falls inward, with

t

continuing to increase, until it crosses the gravitational radius again

r = 2 M at t = + \infty, τ = + {(\frac{r_{max}^{3}}{8 M})}^{1 / 2} \cos^{- 1} (\frac{4 M}{r_{max}} - 1) + r_{max} {(1 - \frac{2 M}{r_{max}})}^{1 / 2}

(event

F^{'}

in diagram);
and it finally falls on in with decreasing

t

(but, of course, still increasing

τ

) to

r = 0 at t = + 2 π M {(\frac{r_{max}}{2 M} - 1)}^{1 / 2} (\frac{r_{max}}{4 M} + 1), τ = + \frac{π}{2} {(\frac{r_{max}^{3}}{2 M})}^{1 / 2}

(event

F^{''}

in diagram).
spread them out into a line in a new (

r_{new}, t_{new}

)-plane; and will squeeze the "line"

(r = 2 M, t

from

- \infty

+ \infty)

into a single point in the (

r_{new}, t_{new})

-plane. One is the more prepared to accept this tentative conclusion and act on it because one has already seen (equation 31.8) that the region covering the

(θ, ϕ) 2

-sphere at

r = 2 M

, and extending from

t = - \infty

t = + \infty

, has zero proper volume. What timelier indication could one want that the "line"

r = 2 M, - \infty < t < \infty

, is actually a point?

§31.4. SEVERAL WELL-BEHAVED COORDINATE SYSTEMS

The well-behaved coordinate system that is easiest to visualize is one in which the radially moving test particles of equations (31.10) remain always at rest ("comoving

Novikov coordinates:
(1) how constructed
(2) line element
coordinates"). Such coordinates were first used by Novikov (1963). Novikov attaches a specific value of his radial coordinate,

R^{*}

, to each test particle as it emerges from the singularity of infinite tidal forces at

r = 0

, and insists that the particle carry that value of

R^{*}

throughout its "cycloidal life"-up through

r = 2 M

r = r_{max}

, then back down through

r = 2 M

r = 0

. For definiteness, Novikov expresses the

R^{*}

value for each particle in terms of the peak point on its trajectory by

\begin{matrix} (31.11) & R^{*} = {(r_{max} / 2 M - 1)}^{1 / 2} \end{matrix}

As a time coordinate, Novikov uses proper time

τ

of the test particles, normalized so

τ = 0

at the peak of the orbit. Every particle in the swarm is ejected in such a manner that it arrives at the summit of its trajectory

(r = r_{max}, τ = 0)

at one and the same value of the Schwarzschild coordinate time; namely, at

t = 0

Simple though they may be conceptually, the Novikov coordinates are related to the original Schwarzschild coordinates by a very complicated transformation: (1) combine equations (31.10b) and (31.11) to obtain

η (τ, R^{*})

; (2) combine

η (τ, R^{*})

with (31.10a) and (31.11) to obtain

r (τ, R^{*})

; (3) combine

η (τ, R^{*})

with (31.10c) and (31.11) to obtain

t (τ, R^{*})

. The resulting coordinate transformation, when applied to the Schwarzschild metric (31.1), yields the line element

\begin{matrix} (31.12a) & d s^{2} = - d τ^{2} + (\frac{R^{* 2} + 1}{R^{* 2}}) {(\frac{\partial r}{\partial R^{*}})}^{2} d R^{* 2} + r^{2} (d θ^{2} + \sin^{2} θ d ϕ^{2}) \end{matrix}

("Schwarzschild geometry in Novikov coordinates".) Here

r

is no longer a radial coordinate; it is now a metric function

r (τ, R^{*})

given implicitly by

\begin{aligned} \frac{τ}{2 M} = \pm (R^{* 2} + 1) {[\frac{r}{2 M} - \frac{(r / 2 M)^{2}}{R^{* 2} + 1}]}^{1 / 2} \\ (31.12b) & + {(R^{* 2} + 1)}^{3 / 2} \cos^{- 1} [{(\frac{r / 2 M}{R^{* 2} + 1})}^{1 / 2}] \end{aligned}

Figure 31.2 shows the locations of several key regions of Schwarzschild spacetime in this coordinate system. The existence of two distinct regions with

r = 0

(singularities) and two distinct regions with

r ⟶ \infty

(asymptotically flat regions; recall that

4 π r^{2} =

surface area!) will be discussed in

§ 31.5

Figure 31.2.

The Novikov coordinate system for Schwarzschild spacetime (schematic). The dashed curves are curves of constant

r

(recall:

4 π r^{2} =

surface area about center of symmetry). The region shaded gray is not part of spacetime; it corresponds to

r < 0

, a region that cannot be reached because of the singularity of spacetime at

r = 0

. Notice that the "line"

(r = 2 M, - \infty <

t < + \infty

) of the Schwarzschild coordinate diagram (Figure 31.1) has been compressed into a point here, in accordance with the discussion at the end of

§ 31.3

Although Novikov's coordinate system is very simple conceptually, the mathematical expressions for the metric components in it are rather unwieldy. Simpler, more usable expressions have been obtained in a different coordinate system ("Kruskal-Szekeres coordinates") by Kruskal (1960), and independently by Szekeres (1960).

Kruskal and Szekeres use a dimensionless radial coordinate

u

and a dimensionless Kruskal-Szekeres coordinates time coordinate

v

related to the Schwarzschild

r

and

t

\begin{matrix} (31.13b) & \begin{array}{l} u = (r / 2 M - 1)^{1 / 2} e^{r / 4 M} \cosh (t / 4 M) \\ v = (r / 2 M - 1)^{1 / 2} e^{r / 4 M} \sinh (t / 4 M) \end{array}} when r > 2 M, \end{matrix}

(Motivation for introducing such coordinates is given in Box 31.2.) By making this change of coordinates in the Schwarzschild metric (31.1), one obtains the following line element:

\begin{matrix} (31.14a) & d s^{2} = (32 M^{3} / r) e^{- r / 2 M} (- d v^{2} + d u^{2}) + r^{2} (d θ^{2} + \sin^{2} θ d ϕ^{2}) \end{matrix}

("Schwarzschild geometry in Kruskal-Szekeres coordinates"). Here

r

is to be regarded as a function of

u

and

v

defined implicitly by

\begin{matrix} (31.14b) & (r / 2 M - 1) e^{r / 2 M} = u^{2} - v^{2} \end{matrix}

[cf. equations (31.13)].

Box 31.2 MOTIVATION FOR KRUSKAL-SZEKERES COORDINATES*

A. EDDINGTON-FINKELSTEIN COORDINATES

The motivation for the Kruskal-Szekeres system begins by introducing a different coordinate system, first devised by Eddington (1924) and rediscovered by Finkelstein (1958). Eddington and Finkelstein use as the foundation of their coordinate system, not freely falling particles as did Novikov, but freely falling photons. More particularly, they introduce coordinates

\tilde{U}

and

\tilde{V}

, which are labels for outgoing and ingoing, radial, null geodesics. The geodesics are given by

d s^{2} = 0 = - (1 - 2 M / r) d t^{2} + (1 - 2 M / r)^{- 1} d r^{2} .

Equivalently, outgoing geodesics are given by

\tilde{U} =

const, where

\begin{matrix} (1a) & \tilde{U} \equiv t - r^{*}; \end{matrix}

and ingoing geodesics are given by

\tilde{V} =

const, where

\begin{matrix} (1b) & \tilde{V} \equiv t + r^{*} \end{matrix}

Here

r^{*}

is the "tortoise coordinate" of

§ 25.5

and Figure 25.4:

\begin{matrix} (Ic) & r^{*} \equiv r + 2 M \ln | r / 2 M - 1 | . \end{matrix}

Ingoing Eddington-Finkelstein Coordinates-Adopt $r$ and $\tilde{V}$ as coordinates in place of $r$ and $t$

The Schwarzschild metric becomes.

\begin{matrix} (2) & d s^{2} = - (1 - 2 M / r) d {\tilde{V}}^{2} + 2 d \tilde{V} d r + r^{2} d Ω^{2} \end{matrix}

The radial light cone,

d s^{2} = 0

, has one leg

\begin{matrix} (3a) & d \tilde{V} / d r = 0, \end{matrix}

and the other leg

\begin{matrix} (3b) & \frac{d \tilde{V}}{d r} = \frac{2}{1 - 2 M / r} \end{matrix}

From this, and this alone, one can infer all features of the drawing.

Ingoing Eddington-Finkelstein coordinates (one rotational degree of freedom is suppressed; i.e.,

θ

is set equal to

π / 2

). Surfaces of constant

\tilde{V}

, being ingoing null surfaces, are plotted on a 45 -degree slant, just as they would be in flat spacetime. Equivalently, surfaces of constant

\tilde{t} \equiv \tilde{V} - r = t + 2 M \ln | r / 2 M - 1 |

are plotted as horizontal surfaces.

Outgoing Eddington-Finkelstein Coordinates-Adopt $r$ and $\tilde{U}$ as coordinates in place of $r$ and $t$

The Schwarzschild metric becomes

\begin{matrix} (4) & d s^{2} = - (1 - 2 M / r) d {\tilde{U}}^{2} - 2 d \tilde{U} d r + r^{2} d Ω^{2} \end{matrix}

Box 31.2 (continued)

The radial light cone,

d s^{2} = 0

, has one leg

\begin{matrix} (5a) & d \tilde{U} / d r = 0, \end{matrix}

and the other leg

\begin{matrix} (5b) & \frac{d \tilde{U}}{d r} = - \frac{2}{1 - 2 M / r} . \end{matrix}

From this, and this alone, one can infer all features of the drawing.

Outgoing Eddington-Finkelstein coordinates (one rotational degree of freedom is suppressed). (Surfaces of constant

\hat{U}

, being outgoing null surfaces, are plotted on a 45 -degree slant, just as they would be in flat spacetime.)

Notice that both Eddington-Finkelstein coordinate systems are better behaved at the gravitational radius than is the Schwarzschild coordinate system; but they are not fully well-behaved. The outgoing coordinates (

\tilde{U}, r, θ, ϕ

) describe in a nonpathological manner the ejection of particles outward from

r = 0

through

r = 2 M

; but their description of infall through

r = 2 M

has the same pathology as the description given by Schwarzschild coordinates (Figure 31.1). Similarly, the ingoing coordinates (

\tilde{V}, r, θ, ϕ

) describe well the infall of a particle through

r = 2 M

, but they give a pathological description of outgoing trajectories. Moreover, the contrast between the two diagrams seems paradoxical: in one the gravitational radius is made up of world lines of outgoing photons; in the other it is made up of world lines of ingoing photons! To resolve the paradox, one must seek another, better-behaved coordinate system. [But note: because the ingoing Eddington-Finklestein coordinates describe infall so well, they are used extensively in discussions of gravitational collapse (Chapter 32) and black holes (Chapters 33 and 34).]

B. TRANSITION FROM EDDINGTON-FINKELSTEIN TO KRUSKAL-SZEKERES

Perhaps one would obtain a fully well-behaved coordinate system by dropping

r

from view and using

\tilde{U}, \tilde{ν}

, as the two coordinates in the radial-time plane. The resulting coordinate system is related to Schwarzschild coordinates by [see equations (1)]

\begin{aligned} (6a) & \tilde{V} - \tilde{U} = 2 r^{*} \\ (6b) & \tilde{V} + \tilde{U} = 2 t \end{aligned}

and the line element in terms of the new coordinates reads

\begin{matrix} (7) & d s^{2} = - (1 - 2 M / r) d \tilde{U} d \tilde{V} + r^{2} (d θ^{2} + \sin^{2} θ d ϕ^{2}) \end{matrix}

Contrary to one's hopes, this coordinate system is pathological at

r = 2 M

.
Second thoughts about the construction reveal the trouble: the surfaces

\tilde{U} =

constant (outgoing null surfaces) used in constructing it are geometrically welldefined, as are the surfaces

\tilde{V} =

constant (ingoing null surfaces); but the way of labeling them is not. Any relabeling,

\tilde{u} = F (\tilde{U})

and

\tilde{v} = G (\tilde{V})

, will leave the surfaces unchanged physically. What one needs is a relabeling that will get rid of the singular factor

1 - 2 M / r

in the line element (7). A successful relabeling is suggested by the equation

\begin{matrix} (8) & \exp [(\tilde{V} - \tilde{U}) / 4 M] = \exp (r^{*} / 2 M) = (r / 2 M - 1) \exp (r / 2 M) \end{matrix}

Box 31.2 (continued)

which follows from equations (6a) and (1c). Experimenting with this relation quickly reveals that the relabeling

\begin{aligned} (9a) & \tilde{u} \equiv - e^{- \tilde{U} / 4 M} = - (r / 2 M - 1)^{1 / 2} e^{r / 4 M} e^{- t / 4 M} \\ (9b) & \tilde{v} \equiv e^{+ \tilde{V} / 4 M} = (r / 2 M - 1)^{1 / 2} e^{r / 4 M} e^{t / 4 M} \end{aligned}

will remove the offending

1 - 2 M / r

from the metric coefficients. In terms of these new coordinates, the line element reads

\begin{matrix} (10a) & d s^{2} = - (32 M^{3} / r) e^{- r / 2 M} d \tilde{v} d \tilde{u} + r^{2} (d θ^{2} + \sin^{2} θ d ϕ^{2}) \end{matrix}

Here

r

is still defined by

4 π r^{2} =

surface area, but it must be regarded as a function of

\tilde{v}

and

\tilde{u}

\begin{matrix} (10b) & (r / 2 M - 1) e^{r / 2 M} = - \tilde{u} \tilde{v} \end{matrix}

One can readily verify that this equation determines

r

uniquely (recall:

r > 0

!) in terms of the product

\tilde{u} \tilde{v}

[details in Misner (1969a)].

The coordinates,

\tilde{u}, \tilde{v}

, which label the ingoing and outgoing null surfaces, are null coordinates; i.e.,

\partial / \partial \tilde{u} \cdot \partial / \partial \tilde{u} = g_{\tilde{u}} \tilde{u} = 0, \partial / \partial \tilde{v} \cdot \partial / \partial \tilde{v} = g_{\tilde{v}} \tilde{v} = 0

[see equation (10a)]. If one is not accustomed to working with null coordinates, it is helpful to replace

\tilde{u}

and

\tilde{v}

by spacelike and timelike coordinates,

u

and

v

(KruskalSzekeres coordinates!) defined by

\begin{aligned} (11a) & u \equiv \frac{1}{2} (\tilde{v} - \tilde{u}) = (r / 2 M - 1)^{1 / 2} e^{r / 4 M} \cosh (t / 4 M) \\ (11b) & v \equiv \frac{1}{2} (\tilde{v} + \tilde{u}) = (r / 2 M - 1)^{1 / 2} e^{r / 4 M} \sinh (t / 4 M) \end{aligned}

so that

\begin{matrix} (12) & d v^{2} - d u^{2} = d \tilde{v} d \tilde{u} \end{matrix}

In terms of these coordinates, the line element has the Kruskal form (31.14), which is fully well-behaved at the gravitational radius.

Although the Kruskal-Szekeres line element is well behaved at

r = 2 M

, the transformation (11) from Schwarzschild to Kruskal-Szekeres is not; it becomes meaningless (

u

and

v

"imaginary") when one moves from

r > 2 M

r < 2 M

. Of course, this is a manifestation of the pathologies of Schwarzschild coordinates. By trial and error, one readily finds a new transformation, to replace (11) at

r < 2 M

, leading from Schwarzschild to Kruskal-Szekeres coordinates:

\begin{aligned} (11c) & u = (1 - r / 2 M)^{1 / 2} e^{r / 4 M} \sinh (t / 4 M) \\ (11d) & v = (1 - r / 2 M)^{1 / 2} e^{r / 4 M} \cosh (t / 4 M) \end{aligned}

§31.5. RELATIONSHIP BETWEEN KRUSKAL-SZEKERES COORDINATES AND SCHWARZSCHILD COORDINATES

In the Kruskal-Szekeres coordinate system, the singularity

r = 0

is located at

v^{2} - u^{2} = 1

. Thus there are actually two singularities, not one; both

\begin{matrix} (31.15) & v = + {(1 + u^{2})}^{1 / 2} and v = - {(1 + u^{2})}^{1 / 2} correspond to r = 0! \end{matrix}

This is not the only surprise that lies hidden in the Kruskal-Szekeres line element (31.14). Notice also that

r ≫ 2 M

(the region of spacetime far outside the gravitational radius) is given by

u^{2} ≫ v^{2}

. Thus there are actually two exterior regions*; both

\begin{matrix} (31.16) & u ≫ + | v | and u ≪ - | v | correspond to r ≫ 2 M! \end{matrix}

How can this be? When the geometry is charted in Schwarzschild coordinates, it contains one singularity and one exterior region; but when expressed in KruskalSzekeres coordinates, it shows two of each. The answer must be that the Schwarzschild coordinates cover only part of the spacetime manifold; they must be only a local coordinate patch on the full manifold. Somehow, by means of the coordinate transformation that leads to Kruskal-Szekeres coordinates, one has analytically extended the limited Schwarzschild solution for the metric to cover all (or more nearly all) of the manifold.

To understand this covering more clearly, transform back from Kruskal-Szekeres coordinates to Schwarzschild coordinates (see Figure 31.3). The transformation equations, as written down in (31.13) were valid only for the quadrants

u > | v |

[equation (31.13a)] and

v > | u |

[equation (31.13b)] of Kruskal coordinates. Denote these quadrants by the numerals I and II; and denote the other quadrants by III and IV (see Figure 31.3). In the other quadrants, one can also transform the KruskalSzekeres line element (31.14) into the Schwarzschild line element (31.1); but slightly different transformation equations are needed. One easily verifies that the following sets of transformations work:

\begin{aligned} (31.17a) & (I) {\begin{matrix} u = (r / 2 M - 1)^{1 / 2} e^{r / 4 M} \cosh (t / 4 M) \\ v = (r / 2 M - 1)^{1 / 2} e^{r / 4 M} \sinh (t / 4 M) \end{matrix}, \\ (31.17b) & (II) {\begin{matrix} u = (1 - r / 2 M)^{1 / 2} e^{r / 4 M} \sinh (t / 4 M) \\ v = (1 - r / 2 M)^{1 / 2} e^{r / 4 M} \cosh (t / 4 M) \end{matrix}, \\ (31.17c) & (III) {\begin{matrix} u = - (r / 2 M - 1)^{1 / 2} e^{r / 4 M} \cosh (t / 4 M) \\ v = - (r / 2 M - 1)^{1 / 2} e^{r / 4 M} \sinh (t / 4 M) \end{matrix}, \\ (31.17~d) & (IV) {\begin{matrix} u = - (1 - r / 2 M)^{1 / 2} e^{r / 4 M} \sinh (t / 4 M) \\ v = - (1 - r / 2 M)^{1 / 2} e^{r / 4 M} \cosh (t / 4 M) \end{matrix} . \end{aligned}

Kruskal-Szekeres coordinates reveal that Schwarzschild spacetime has two "

r = 0

singularities" and two "

r ⟶ \infty

exterior regions"

Transformation between Schwarzschild coordinates and Kruskal-Szekeres coordinates

Figure 31.3.
The transformation of the Schwarzschild vacuum geometry between Schwarzschild and Kruskal-Szekeres coordinates. Two Schwarzschild coordinate patches I, II, and III, IV (illustrated in the upper and lower portions of Figure 31.5 ,a) are required to cover the complete Schwarzschild geometry, whereas a single Kruskal-Szekeres coordinate system suffices. The Schwarzschild geometry consists of four regions I, II, III, IV. Regions I and III represent two distinct, but identical, asymptotically flat universes in which

r > 2 M

; while regions II and IV are two identical, but time-reversed, regions in which physical singularities

(r = 0)

evolve. The transformation laws that relate the Schwarzschild and Kruskal-Szekeres coordinate systems to each other are given by equations (31.17) and (31.18). In the Kruskal-Szekeres

u, v

-plane, curves of constant

r

are hyperbolae with asymptotes

u = \pm v

, while curves of constant

t

are straight lines through the origin.

The inverse transformations are

\begin{matrix} (31.18a) & (r / 2 M - 1) e^{r / 2 M} = u^{2} - v^{2} in I, II, III, IV; \\ (31.18b) & t = {\begin{cases} 4 M \tanh^{- 1} (v / u) in I and III, \\ 4 M \tanh^{- 1} (u / v) in II and IV. \end{cases} \end{matrix}

These coordinate transformations are exhibited graphically in Figure 31.3. Notice that two Schwarzschild coordinate patches, I, II, and III, IV, are required to cover the entire Schwarzschild geometry; but a single Kruskal coordinate system suffices. Schwarzschild patch I, II, is divided into two regions-region I, which is outside the gravitational radius

(r > 2 M)

, and region II, which is inside the gravitational radius (

r < 2 M

). Similarly, Schwarzschild patch III, IV, consists of an exterior region (III) and an interior region (IV).

Two Schwarzschild coordinate patches are required to cover all of spacetime

Figure 31.4.
(a) Typical radial timelike

(A, E, F)

, lightlike

(B)

, and spacelike

(C, D)

geodesics of the Schwarzschild geometry, as seen in the Schwarzschild coordinate system (schematic only). This is a reproduction of Figure 31.1.
(b) The same geodesics, as seen in the Kruskal-Szekeres coordinate system, and as extended either to infinite length or to the singularity of infinite curvature at

r = 0

(schematic only).

Equations (31.18) reveal that the regions of constant

r

(constant surface area) are hyperbolae with asymptotes

u = \pm v

in the Kruskal-Szekeres diagram, and that regions of constant

t

are straight lines through the origin.

Several radial geodesics of the complete Schwarzschild geometry are depicted in the Kruskal-Szekeres coordinate system in Figure 31.4. Notice how much more reasonable the geodesic curves look in Kruskal-Szekeres coordinates than in Schwarzschild coordinates. Notice also that radial, lightlike geodesics (paths of radial light rays) are 45-degree lines in the Kruskal-Szekeres coordinate system. This can be seen from the Kruskal-Szekeres line element (31.14), for which

d u = \pm d v

guarantees

d s = 0

. Because of this 45 -degree property, the radial light cone in a Kruskal-Szekeres diagram has the same form as in the space-time diagram of special relativity. Any radial curve that points "generally upward" (i.e., makes an angle of less than 45 degrees with the vertical,

v

, axis) is timelike; and curves that point "generally outward" are spacelike. This property enables a Kruskal-Szekeres diagram to exhibit easily the causality relation between one event in spacetime and another (see exercises 31.2 to 31.4 ).

Properties of the Kruskal-Szekeres coordinate system

EXERCISES

Exercise 31.2. NONRADIAL LIGHT CONES

Show that the world line of a photon traveling nonradially makes an angle less than 45 degrees with the vertical

v

-axis of a Kruskal-Szekeres coordinate diagram. From this, infer that particles with finite rest mass, traveling nonradially or radially, must always move "generally upward" (angle less than 45 degrees with vertical

v

-axis).

Exercise 31.3. THE CRACK OF DOOM

Use a Kruskal diagram to show the following.
(a) If a man allows himself to fall through the gravitational radius

r = 2 M

, there is no way whatsoever for him to avoid hitting (and being killed in) the singularity at

r = 0

.
(b) Once a man has fallen inward through

r = 2 M

, there is no way whatsoever that he can send messages out to his friends at

r > 2 M

, but he can still receive messages from them (e.g., by radio waves, or laser beam, or infalling "CARE packages").

Exercise 31.4. HOW LONG TO LIVE?

Show that once a man falling inward reaches the gravitational radius, no matter what he does subsequently (no matter in what directions, how long, and how hard he blasts his rocket engines), he will be pulled into the singularity and killed in a proper time of

\begin{matrix} (31.19) & τ < τ_{max} = π M = 1.54 \times 10^{- 5} (M / M_{⊙}) seconds. \end{matrix}

[Hint: The trajectory of longest proper time lapse must be a geodesic. Use the mathematical tools of Chapter 25 to show that the geodesic of longest proper time lapse between

r = 2 M

and

r = 0

is the radial geodesic (31.10a), with

r_{max} = 2 M

, for which the time lapse is

π M

Exercise 31.5. EDDINGTON-FINKELSTEIN AND KRUSKAL-SZEKERES COMPARED

Use coordinate diagrams to compare the ingoing and outgoing Eddington-Finkelstein coordinates of Box 31.2 with the Kruskal-Szekeres coordinates. Pattern the comparison after that between Schwarzschild and Kruskal-Szekeres in Figures 31.3 and 31.4.

Exercise 31.6. ANOTHER COORDINATE SYSTEM

Construct a coordinate diagram for the

\tilde{U}, \tilde{V}, θ, ϕ

coordinate system of Box 31.2 [equations (6) and (7)]. Show such features as (1) the relationship to Schwarzschild and to KruskalSzekeres coordinates; (2) the location of

r = 2 M

; and (3) radial geodesics.

§31.6. DYNAMICS OF THE SCHWARZSCHILD GEOMETRY

What does the Schwarzschild geometry look like? This question is most readily answered by means of embedding diagrams analogous to those for an equilibrium star (

$ 23.8

; Figure 23.1; and end of Box 23.2) and for Friedmann universes of positive and negative spatial curvature [equations (27.23) and (27.24) and Box 27.2].

Examine, first, the geometry of the spacelike hypersurface

v = 0

, which extends from

u = + \infty (r = \infty)

into

u = 0 (r = 2 M)

and then out to

u = - \infty (r = \infty)

. In Schwarzschild coordinates this surface is a slice of constant time,

t = 0

[see equation (31.18b)]; it is precisely the surface for which an embedding diagram was calculated in equation (23.34b). The embedded surface, with one degree of rotational freedom suppressed, is described by the paraboloid of revolution

\begin{matrix} (31.20) & \bar{r} = 2 M + {\bar{z}}^{2} / 8 M \end{matrix}

Figure 31.5.
(a) The Schwarzschild space geometry at the "moment of time"

t = v = 0

, with one degree of rotational freedom suppressed

(θ = π / 2)

. To restore that rotational freedom and obtain the full Schwarzschild 3-geometry, one mentally replaces the circles of constant

\bar{r} = {({\bar{x}}^{2} + {\bar{y}}^{2})}^{1 / 2}

with spherical surfaces of area

4 π {\bar{r}}^{2}

. Note that the resultant 3 -geometry becomes flat (Euclidean) far from the throat of the bridge in both directions (both "universes").
(b) An embedding of the Schwarzschild space geometry at "time"

t = v = 0

, which is geometrically identical to the embedding (a), but which is topologically different. Einstein's field equations fix the local geometry of spacetime, but they do not fix its topology; see the discussion at end of Box 27.2 . Here the Schwarzschild "wormhole" connects two distant regions of a single, asymptotically flat universe. For a discussion of issues of causality associated with this choice of topology, see Fuller and Wheeler (1962).
in the flat Euclidean space with metric

\begin{matrix} (31.21) & d σ^{2} = d {\bar{r}}^{2} + d {\bar{z}}^{2} + {\bar{r}}^{2} d {\bar{ϕ}}^{2} \end{matrix}

(See Figure 31.5.)
Notice from the embedding diagram of Figure 31.5,a, that the Schwarzschild

The 3-surface

v = t = 0

is a "wormhole" connecting two asymptotically flat universes, or two different regions of one universe

Schwarzschild geometry is dynamic in regions

r < 2 M

Time evolution of the wormhole: creation; expansion; recontraction; and pinch-off

Communication through the wormhole is impossible: it pinches off too fast
geometry on the spacelike hypersurface

t =

const consists of a bridge or "wormhole" connecting two distinct, but identical, asymptotically flat universes. This bridge is sometimes called the "Einstein-Rosen bridge" and sometimes the "Schwarzschild throat" or the "Schwarzschild wormhole." If one so wishes, one can change the topology of the Schwarzschild geometry by connecting the two asymptotically flat universes together in a region distant from the Schwarzschild throat [Fuller and Wheeler (1962); Fig. 31.5b]. The single, unique universe then becomes multiply connected, with the Schwarzschild throat providing one spacelike path from point

a

to point

B

, and the nearly flat universe providing another. For concreteness, focus attention on the interpretation of the Schwarzschild geometry, not in terms of Wheeler's multiply connected single universe, but rather in terms of the EinsteinRosen double universe of Figure 31.5,a.

One is usually accustomed to think of the Schwarzschild geometry as static. However, the static "time translations,"

t ⟶ t + Δ t

, which leave the Schwarzschild geometry unchanged, are time translations in the strict sense of the words only in regions I and III of the Schwarzschild geometry. In regions II and IV,

t ⟶ t + Δ t

is a spacelike motion, not a timelike motion (see Fig. 31.3). Consequently, a spacelike hypersurface, such as the surface

t =

const of Figure 31.5,a, which extends from region I through

u = v = 0

into region III, is not static. As this spacelike hypersurface is pushed forward in time (in the

+ v

direction of the Kruskal diagram), it enters region II, and its geometry begins to change.
In order to examine the time-development of the Schwarzschild geometry, one needs a sequence of embedding diagrams, each corresponding to the geometry of a spacelike hypersurface to the future of the preceding one. But how are the hypersurfaces to be chosen? In Newtonian theory or special relativity, one chooses hypersurfaces of constant time. But in dynamic regions of curved spacetime, no naturally preferred time coordinate exists. This situation forces one to make a totally arbitrary choice of hypersurfaces to use in visualizing the time-development of geometry, and to keep in mind how very arbitrary that choice was.

Figure 31.6 uses two very different choices of hypersurfaces to depict the timedevelopment of the Schwarzschild geometry. (Still other choices are shown in Figure 21.4.) Notice that the precise geometry of the evolving bridge depends on the arbitrary choice of spacelike hypersurfaces, but that the qualitative nature of the evolution is independent of the choice of hypersurfaces. Qualitatively speaking, the two asymptotically flat universes begin disconnected, with each one containing a singularity of infinite curvature

(r = 0)

. As the two universes evolve in time, their singularities join each other and form a nonsingular bridge. The bridge enlarges, until it reaches a maximum radius at the throat of

r = 2 M

(maximum circumference of

4 π M

; maximum surface area of

16 π M^{2}

). It then contracts and pinches off, leaving the two universes disconnected and containing singularities

(r = 0)

once again. The formation, expansion, and collapse of the bridge occur so rapidly, that no particle or light ray can pass across the bridge from the faraway region of the one universe to the faraway region of the other without getting caught and crushed in the throat as it pinches off. (To verify this, examine the Kruskal-Szekeres diagram of Figure 31.3, where radial light rays move along 45 -degree lines.)

Spacelike slices

History

A - B - C - D - E - F - G

History

A - W - X - D - Y - Z - G

Figure 31.6.
Dynamical evolution of the Einstein-Rosen bridge of the vacuum Schwarzschild geometry (schematic). Shown here are two sequences of embedding diagrams corresponding to two different ways of viewing the evolution of the bridge-History

A - B - C - D - E - F - G

, and History

A - W - X - D - Y - Z - G

. The embedding diagrams are skeletonized in that each diagram must be rotated about the appropriate vertical axis in order to become two-dimensional surfaces analogous to Figure 31.5 , a. [Notice that the hypersurfaces of which embedding diagrams are given intersect the singularity only tangentially. Hypersurfaces that intersect the singularity at a finite angle in the

u, v

-plane are not shown because they cannot be embedded in a Euclidean space. Instead, a Minkowski space (indefinite metric) must be used, at least near

r = 0

. For an example of an embedding in Minkowski space, see the discussion of a universe with constant negative spatial curvature in equations (27.23) and (27.24) and Box 27.2C.] Figure 21.4 exhibits embedding diagrams for other spacelike slices in the Schwarzschild geometry.

From the Kruskal-Szekeres diagram and the 45 -degree nature of its radial light rays, one sees that any particle that ever finds itself in region IV of spacetime must have been "created" in the earlier singularity; and any particle that ever falls into region II is doomed to be crushed in the later singularity. Only particles that stay forever in one of the asymptotically flat universes I or III, outside the gravitational radius

(r > 2 M)

, are forever safe from the singularities.

Some investigators, disturbed by the singularities at

r = 0

or by the "double-universe" nature of the Schwarzschild geometry, have proposed modifications of its topology. One proposal is that the earlier and later singularities be identified with each other, so that a particle which falls into the singularity of region II, instead of being destroyed, will suddenly reemerge, being ejected, from the singularity of region IV. One cannot overstate the objections to this viewpoint: the region

r = 0

is a physical singularity of infinite tidal gravitation forces and infinite Riemann curvature. Any particle that falls into that singularity must be destroyed by those

Creation and destruction in the singularities

Nonviable proposals for modifying the topology of Schwarzschild spacetime
forces. Any attempt to extrapolate its fate through the singularity using Einstein's field equations must fail; the equations lose their predictive power in the face of infinite curvature. Consequently, to postulate that the particle reemerges from the earlier singularity is to make up an ad hoc mathematical rule, one unrelated to physics. It is conceivable, but few believe it true, that any object of finite mass will modify the geometry of the singularity as it approaches

r = 0

to such an extent that it can pass through and reemerge. However, whether such a speculation is correct must be answered not by ad hoc rules, but by concrete, difficult computations within the framework of general relativity theory (see Chapter 34).
A second proposal for modifying the topology of the Schwarzschild geometry is this: one should avoid the existence of two different asymptotically flat universes by identifying each point

(v, u, θ, ϕ)

with its opposite point

(- v, - u, θ, ϕ)

in the Kruskal-Szekeres coordinate system. Two objections to this proposal are: (1) it produces a sort of "conical" singularity (absence of local Lorentz frames) at

(v, u) =

(0, 0)

, i.e., at the neck of the bridge at its moment of maximum expansion; and (2) it leads to causality violations in which a man can meet himself going backward in time.

One good way for the reader to become conversant with the basic features of the Schwarzschild geometry is to reread

$ $ 31.1 - 31.4

carefully, reinterpreting everything said there in terms of the Kruskal-Szekeres diagram.

EXERCISES

Exercise 31.7. SCHWARZSCHILD METRIC IN ISOTROPIC COORDINATES

(a) Show that, rewritten in the isotropic coordinates of Exercise 23.1, the Schwarzschild metric reads

\begin{matrix} (31.22) & d s^{2} = - {(\frac{1 - M / 2 \bar{r}}{1 + M / 2 \bar{r}})}^{2} d t^{2} + {(1 + \frac{M}{2 \bar{r}})}^{4} [d {\bar{r}}^{2} + {\bar{r}}^{2} (d θ^{2} + \sin^{2} θ d ϕ^{2})]; \end{matrix}

and derive the transformation

\begin{matrix} (31.23) & r = \bar{r} (1 + M / 2 \bar{r})^{2} \end{matrix}

between the two radial coordinates.
(b) Which regions of spacetime (I, II, III, IV; see Figure 31.3) are covered by the isotropic coordinate patch, and which are not?
(c) Calculate and construct an embedding diagram for the spacelike hypersurface

t = 0

0 < \bar{r} < \infty

.
(d) Find a coordinate transformation that interchanges the region near

\bar{r} = 0

with the region near

\bar{r} = \infty

, while leaving the metric coefficients in their original form.

Exercise 31.8. REISSNER-NORDSTR $\emptyset$ M GEOMETRY

(a) Solve the Einstein field equations for a spherically symmetric, static gravitational field

d s^{2} = - e^{2 ϕ (r)} d t^{2} + e^{2 A (r)} d r^{2} + r^{2} (d θ^{2} + \sin^{2} θ d ϕ^{2}),

with no matter present, but with a radial electric field

B = 0, E = f (r) e_{\hat{r}}

in the static orthonormal frame

ω^{\hat{t}} = e^{ϕ} d t, ω^{\hat{ı}} = e^{\hat{4}} d r, ω^{\hat{θ}} = r d θ, ω^{\hat{ϕ}} = r \sin θ d ϕ .

Use as a source in the Einstein field equations the stress-energy of the electric field. [Answer:

\begin{matrix} (31.24a) & E = (Q / r^{2}) e_{\hat{r}} \\ (31.24b) & d s^{2} = - (1 - \frac{2 M}{r} + \frac{Q^{2}}{r^{2}}) d t^{2} + {(1 - \frac{2 M}{r} + \frac{Q^{2}}{r^{2}})}^{- 1} d r^{2} + r^{2} (d θ^{2} + \sin^{2} θ d ϕ^{2}) \end{matrix}

This is called the "Reissner (1916)-Nordstrom (1918) metric".]
(b) Show that the constant

Q

is the total charge as measured by a distant observer

(r ⩾ 2 M

and

r ≫ Q

), who uses a Gaussian flux integral, or who studies the coulomb-force-dominated orbits of test charges with charge-to-mass ratio

e / μ ⩾ M / Q

. What is the charge-to-mass ratio, in dimensionless units, for an electron? Show that the constant

M

is the total mass as measured by a distant observer using the Keplerian orbits of electrically neutral particles.
(c) Show that for

Q > M

, the Reissner-Nordstrom coordinate system is well-behaved from

r = \infty

down to

r = 0

, where there is a physical singularity and infinite tidal forces.
(d) Explore the nature of the spacetime geometry for

Q < M

, using all the techniques of this chapter (coordinate transformations, Kruskal-like coordinates, studies of particle orbits, embedding diagrams, etc.).
[Solution: see Graves and Brill (1960); also Fig. 34.4 of this book.]
(e) Similarly explore the spacetime geometry for

Q = M

. [Solution: see Carter (1966b).]
(f) For the case of a large ratio of charge to mass

[Q > M

as in part (c)], show that the region near

r = 0

is unphysical. More precisely, show that any spherically symmetric distribution of charged stressed matter that gives rise to the fields (31.24) outside its boundary must modify these fields for

r < r_{0} = Q^{2} / 2 M

. [Hint: Study the quantity

m (r)

defined in equations (23.18) and (32.22h), noting its values deduced from equation (31.24), on the one hand, and from the appropriate Einstein equation within the matter distribution, on the other hand. See Figure 26 of Misner (1969a) for a similar argument.]

cmorest 32

GRAVITATIONAL COLLAPSE

Now, here, you see, it takes all the running you can do, to keep in the same place. If you want to get somewhere else, you must run twice as fast as that.

The roles and relevance of the Schwarzschild geometry

§32.1. RELEVANCE OF SCHWARZSCHILD GEOMETRY

The story that unfolded in the preceeding chapter was fantastic! One began with the innocuous looking Schwarzschild line element

\begin{matrix} (32.1) & d s^{2} = - (1 - \frac{2 M}{r}) d t^{2} + \frac{d r^{2}}{1 - 2 M / r} + r^{2} (d θ^{2} + \sin^{2} θ d ϕ^{2}) \end{matrix}

which was derived originally as the external field of a static star. One asked what happens if the star is absent; i.e., one probed the nature of the Schwarzschild geometry when no star is present to generate it. One might have expected the geometry to be that of a point mass sitting at

r = 0

. But it was not. It turned out to represent a "wormhole" connecting two asymptotically flat universes. Moreover, the wormhole was dynamic. It was created by the "joining together" of two "

r = 0

" singularities, one in each universe; it expanded to a maximum circumference of

4 π M

; it then recontracted and pinched off, leaving the two universes disconnected once again, each with its own "

r = 0

" singularity.
As a solution to Einstein's field equations, this expanding and recontracting wormhole must be taken seriously. It is an exact solution; and it is one of the simplest of all exact solutions. But there is no reason whatsoever to believe that such wormholes exist in the real universe! They can exist only if the expanding universe,

\sim 10 \times 10^{9}

years ago, was "born" with the necessary initial conditions-with "

r = 0

Schwarzschild singularities ready and waiting to blossom forth into wormholes. There is no reason at all to believe in such pathological initial conditions!

Why, then, was so much time and effort spent in Chapter 31 on understanding the Schwarzschild geometry? (1) Because it illustrates clearly the highly nonEuclidean character of spacetime geometry when gravity becomes strong; (2) because it illustrates many of the techniques one can use to analyze strong gravitational fields; and most importantly (3) because, when appropriately truncated, it is the spacetime geometry of a black hole and of a collapsing star-as well as of a wormhole.

This chapter explores the role of the Schwarzschild geometry in gravitational collapse; the next chapter explores its role in black-hole physics.

§32.2. BIRKHOFF'S THEOREM

That the Schwarzschild geometry is relevant to gravitational collapse follows from Birkhoff's (1923) theorem: Let the geometry of a given region of spacetime (1) be spherically symmetric, and (2) be a solution to the Einstein field equations in vacuum. Then that geometry is necessarily a piece of the Schwarzschild geometry. The external field of any electrically neutral, spherical star satisfies the conditions of Birkhoff's theorem, whether the star is static, vibrating, or collapsing. Therefore the external field must be a piece of the Schwarzschild geometry.

Birkhoff's theorem is easily understood on physical grounds. Consider an equilibrium configuration that is unstable against gravitational collapse and that, like all equilibrium configurations (see

§ 23.6

), has the Schwarzschild geometry as its external gravitational field. Perturb this equilibrium configuration in a spherically symmetric way, so that it begins to collapse radially. The perturbation and subsequent collapse cannot affect the external gravitational field so long as exact spherical symmetry is maintained. Just as Maxwell's laws prohibit monopole electromagnetic waves, so Einstein's laws prohibit monopole gravitational waves. There is no possible way for any gravitational influence of the radial collapse to propagate outward.

Not only is Birkhoff's theorem easy to understand, but it is also fairly easy to prove. Consider a spherical region of spacetime. Spherical symmetry alone is sufficient to guarantee that conditions (i), (ii), and (iii) of Box 23.3 are satisfied, and thus to guarantee that one can introduce Schwarzschild coordinates

\begin{matrix} d s^{2} = - e^{2 ϕ} d t^{2} + e^{2 Λ} d r^{2} + r^{2} (d θ^{2} + \sin^{2} θ d ϕ^{2}), \\ (32.2) & Φ = Φ (t, r), and Λ = Λ (t, r) . \end{matrix}

[See Box 23.3 for proof; and notice that: (1) for generality one must allow

g_{t t} = - e^{2 Φ}

and

g_{r r} = e^{2 Λ}

to be positive or negative (no constraint on sign!); (2) at events where the gradient of the "circumference function"

r

is zero or null, Schwarzschild coordinates cannot be introduced. The special case

(\nabla r)^{2} = 0

is treated in exercise 32.1.]

The uniqueness of the Schwarzschild geometry: Birkhoff's theorem

The physics underlying Birkhoff's theorem

Proof of Birkhoff's theorem

Impose Einstein's vacuum field equation on the metric (32.2), using the orthonormal components of the Einstein tensor as derived in exercise 14.16:

\begin{aligned} (32.3a) & G_{\hat{t} \hat{t}} = & r^{- 2} (1 - e^{- 2 Λ}) + 2 (Λ, r / r) e^{- 2 Λ} = 0, \\ (32.3b) & G_{\hat{t} \hat{r}} = & G_{\hat{r} \hat{t}} = 2 (Λ, t r) e^{- (Λ + ϕ)} = 0, \\ (32.3c) & G_{\hat{r} \hat{r}} = & 2 (Φ_{, r} / r) e^{- 2 Λ} + r^{- 2} (e^{- 2 Λ} - 1) = 0, \\ G_{\hat{θ} \hat{θ} \hat{}} = G_{\hat{ϕ} \hat{ϕ}} = & + (Φ_{, r r} + Φ_{, r}^{2} - Φ Φ_{, r} Λ_{r} + Φ_{, r} / r - Λ,_{r} / r) e^{- 2 Λ} \\ (32.3d) & - (Λ_{, t t} + Λ_{, t}^{2} - Λ_{, t} Φ_{, t}) e^{- 2 ϕ} = 0 . \end{aligned}

Equation (32.3b) guarantees that

Λ

is a function of

r

only, and equation (32.3a) then guarantees that

Λ

has the same form as for the Schwarzschild metric:

\begin{matrix} (32.4a) & Λ = - \frac{1}{2} \ln | 1 - 2 M / r | \end{matrix}

Equations (32.3c,d) then become two equivalent equations for

Φ (t, r)

-equivalent by virtue of the Bianchi identity,

\nabla \cdot G = 0

-whose solution is

\begin{matrix} (32.4b) & Φ = \frac{1}{2} \ln | 1 - 2 M / r | + f (t) \end{matrix}

Here

f

is an arbitrary function. Put expressions (32.4) into the line element (32.2); thereby obtain

d s^{2} = - e^{2 f (t)} (1 - \frac{2 M}{r}) d t^{2} + \frac{d r^{2}}{1 - 2 M / r} + r^{2} (d θ^{2} + \sin^{2} θ d ϕ^{2})

Then redefine the time coordinate

t_{new} = \int e^{f (t)} d t

and thereby bring the line element into the Schwarzschild form

d s^{2} = - (1 - \frac{2 M}{r}) d t^{2} + \frac{d r^{2}}{1 - 2 M / r} + r^{2} (d θ^{2} + \sin^{2} θ d ϕ^{2})

Conclusion: When the spacetime surrounding any object has spherical symmetry and is free of charge, mass, and all fields other than gravity, then one can introduce coordinates in which the metric is that of Schwarzschild. Conclusion restated in coordinate-free language: the geometry of any spherically symmetric vacuum region of spacetime is a piece of the Schwarzschild geometry (Birkhoff's theorem). Q.e.D.

EXERCISE

Exercise 32.1. UNIQUENESS OF REISSNER-NORDSTR $ϕ$ M GEOMETRY [Track 2]

Prove the following generalization of Birkhoff's theorem. Let the geometry of a given region of spacetime (1) be spherically symmetric, and (2) be a solution to the Einstein field equations
with an electromagnetic field as source. Then that geometry is necessarily a piece of the Reissner-Nordstrom geometry [equation (31.24b)] with electric and magnetic fields, as measured in the standard static orthonormal frames

E = (Q_{e} / r^{2}) e_{\hat{r}}, B = (Q_{m} / r^{2}) e_{\hat{r}}, Q = {(Q_{e}^{2} + Q_{m}^{2})}^{1 / 2}

[Hints: (1) First consider regions of spacetime in which

(\nabla r)^{2} \neq 0

, using the same methods as the text uses for Birkhoff's theorem. The result is the Reissner-Nordstrom solution. (2) Any region of dimensionality less than four, in which

(\nabla r)^{2} = 0

(e.g., the Schwarzschild radius), can be treated as the join between four-dimensional regions with

(\nabla r)^{2} \neq 0

. Moreover, the geometry of such a region is determined uniquely by the geometry of the adjoining four-dimensional regions ("junction conditions"; §21.13). Since the adjoining regions are necessarily Reissner-Nordstrøm (step 1), then so are such "sandwiched" regions. (3) Next consider four-dimensional regions in which

\nabla r = d r

is null and nonzero. Show that in such regions there exist coordinate systems with

d s^{2} = - 2 Ψ d r d t + r^{2} (d θ^{2} + \sin^{2} θ d ϕ^{2})

where

Ψ = Ψ (r, t)

. Show further that the Ricci tensor for this line element has an orthonormalized component

R_{\hat{θ} \hat{θ}} = 1 / r^{2}

whereas the stress-energy tensor for a spherically symmetric electromagnetic field has

8 π T_{\hat{θ} \hat{θ}} = 8 π (T_{\hat{θ} \hat{θ}} - \frac{1}{2} g_{\hat{θ} \hat{θ}} T) = Q^{2} / r^{4}, Q = const.

These quantities,

R_{\hat{θ} \hat{θ}}

and

8 π T_{\hat{θ} \hat{θ}}

, must be equal (Einstein's field equation) but cannot be because of their different

r

-dependence. Thus, an electromagnetic field cannot generate regions with

d r \neq 0, d r \cdot d r = 0

. (4) Finally, consider four-dimensional regions in which

d r = 0

. Denote the constant value of

r

a

, and show that any event can be chosen as the origin of a locally well-behaved coordinate system with

\begin{matrix} d s^{2} = a^{2} (- d {\tilde{τ}}^{2} + e^{2 λ} d z^{2} + d θ^{2} + \sin^{2} θ d ϕ^{2}) \\ λ = λ (\tilde{τ}, z), λ (\tilde{τ} = 0, z) = 0, \dot{λ} (\tilde{τ} = 0, z) = 0 . \end{matrix}

[Novikov-type coordinate system; see §31.4.] Show that, in the associated orthonormal frame, spherical symmetry demands

E = (Q_{e} / a^{2}) e_{\hat{z}}, B = (Q_{m} / a^{2}) e_{\hat{z}}, Q \equiv {(Q_{e}^{2} + Q_{m}^{2})}^{1 / 2}

and that the Einstein field equations then require

Q = a

and

e^{λ} = \cos \tilde{τ}

, so that

d s^{2} = Q^{2} (- d {\tilde{τ}}^{2} + \cos^{2} \tilde{τ} d z^{2} + d θ^{2} + \sin^{2} θ d ϕ^{2})

(5) This solution of the field equations [sometimes called the "Bertotti(1959)-Robinson (1959a) Electromagnetic Universe," and explored in this coordinate system by Lindquist (1960)] is actually the throat of the Reissner-Nordstrom solution for the special case

Q = M

. Verify this claim by performing the following coordinate transformation on the Reissner-Nordstrom throat region [equation (31.24b) with

Q = M

and

| r - Q | ≪ Q

r - Q = Q e^{- z} \cos \tilde{τ}, t = Q e^{z} \tan \tilde{τ}

(6) Thus, each possible case leads either to no solution at all, or to a segment of the Reissner-Nordstrom geometry. Q.e.D.] Note: The missing case,

(\nabla r)^{2} = 0

, in the text's proof of Birkhoff's theorem, is resolved by noting that, for

Q = 0

, steps (3) and (4) above lead to no solutions at all. We thank G. F. R. Ellis for pointing out the omission of the case

(\nabla r)^{2} = 0

in the preliminary version of this book.

Gravitational collapse analyzed by examining the star's exterior, Schwarzschild geometry

The gravitational radius as a point of no return, and the "crushing" at

r = 0

§32.3. EXTERIOR GEOMETRY OF A COLLAPSING STAR

Consider a star that is momentarily static, but will subsequently begin to collapse. Its space geometry at the initial moment of Schwarzschild coordinate time,

t = 0

, has two parts: in the exterior, vacuum region

(r > R > 2 M)

, it is the Schwarzschild geometry (Birkhoff's theorem!); but in the star's interior, it is some other, totally different geometry. Whatever the interior geometry may be, it has an embedding diagram at time

t = 0

which is qualitatively like that of Figure 23.1. (For discussion and proof of this, see §23.8.) Notice that the star's space geometry is obtained by discarding the lower universe of the full Schwarzschild geometry (Figure 31.5,a), and replacing it with a smooth "bowl" on which the matter of the star is contained.
To follow the subsequent collapse of this star in the Schwarzschild coordinate system, or in the Kruskal-Szekeres coordinate system, or in an ingoing EddingtonFinkelstein coordinate system, one can similarly discard that part of the coordinate diagram which lies inside the star's surface, and keep only the exterior Schwarzschild region. (See Figure 32.1.) In place of the discarded interior Schwarzschild region, one must introduce some other coordinate system, line element, and diagram that correctly describe the interior of the collapsing star.

From truncated coordinate diagrams (such as Figures 32.1,a,b,c), one can readily discover and understand the various peculiar features of collapse through the gravitational radius.
(1) No matter how stiff may be the matter of which a (spherical) star is made, once its surface has collapsed within the gravitational radius, the star will continue to collapse until its surface gets crushed in the singularity at

r = 0

. This one discovers by recalling that the star's surface cannot move faster than the speed of light, so its world line must always make an angle of less than 45 degrees with the

v

-axis of the Kruskal-Szekeres diagram.
(2) No signal (e.g., photon) emitted from the star's surface after it collapses inside the gravitational radius can ever escape to an external observer. Rather, all signals emitted from inside the gravitational radius get caught and destroyed by the collapse of the surrounding geometry into the singularity at

r = 0

as space "pinches off" around the star.
(3) Consequently, an external observer can never see the star after it passes the gravitational radius; and he can never see the singularity that terminates its col-lapse-unless he chooses to fall through the gravitational radius himself and pay the price of death for the knowledge gained.

Does this mean that the collapsing star instantaneously and completely disappears from external view as it reaches the gravitational radius? No, not according to the analysis depicted in Figure 31.1,c: Place an astrophysicist on the surface of a collapsing star, and have him send a series of uniformly spaced signals to a distant astronomer, at rest at

r ≫ 2 M

, to inform him of the progress of the collapse. These signals propagate along null lines in the spacetime diagram of Figure 31.1c. The signals originate on the world line of the stellar surface, and they are received by the distant astronomer when they intersect his world line,

r =

constant

≫ M

. As the star collapses closer and closer to its gravitational radius,

R = 2 M

, the signals, which are sent at equally spaced intervals according to the astrophysicist's clock, are received by the astronomer at more and more widely spaced intervals. The astronomer does not receive a signal emitted just before the gravitational radius is reached until after an infinite amount of time has elapsed; and he never receives signals emitted after the gravitational radius has been passed. Those signals, like the astrophysicist who sends them, after brief runs get caught and destroyed by the collapsing geometry in the singularity, at

r = 0

. It is not only the star that collapses. The geometry around the star collapses.

Hence, to the distant astronomer, the collapsing star appears to slow down as it approaches its gravitational radius: light from the star becomes more and more red-shifted. Clocks on the star appear to run more and more slowly. It takes an infinite time for the star to reach its gravitational radius; and, as seen by the distant astronomer, the star never gets beyond there.

The optical appearance of a collapsing star was first analyzed mathematically, giving main attention to radially propagating photons, by J. R. Oppenheimer and H. Snyder (1939). More recently a number of workers have reexamined the problem [see, e.g., Podurets (1964), Ames and Thorne (1968) and Jaffe (1969)]. The most important quantitative results of these studies are as follows. In the late stages of collapse, when the distant astronomer sees the star to be very near its gravitational radius, he observes its total luminosity to decay exponentially in time

\begin{matrix} (32.5) & L \propto \exp (- \frac{2}{3 \sqrt{3}} \frac{t}{2 M}) . \end{matrix}

Simultaneously, photons that travel to him along radial trajectories arrive with exponentially increasing redshifts

\begin{matrix} (32.6) & z = Δ λ / λ \propto e^{t / 4 M} \end{matrix}

However, the light from the star is dominated in these late stages, not by photons flying along radial trajectories from near the gravitational radius, but by photons that were deposited by the star in unstable circular orbits as its surface passed through

r = 3 M

(see

$ 25.6

and Box 25.7 ). As time passes, these photons gradually leak out the diffuse spherical shell of trapped photons at

r = 3 M

and fly off to the distant observer, who measures them to have redshift

z \approx 2

. Consequently, in the late stages of collapse the star's spectral lines are broadened enormously, but they are brightest at redshift

z \approx 2

The redshift of signals emitted just before passage through the gravitational radius

Optical appearance of the collapsing star

(a) Schwarzschild

(b) Kruskal-Szekeres

Figure 32.1.
The free-fall collapse of a star of initial radius

R_{i} = 10 M

as depicted alternatively in (a) Schwarzschild coordinates, (b) Kruskal-Szekeres coordinates, and (c) ingoing Eddington-Finkelstein coordinates (see Box 31.2). The region of spacetime inside the collapsing star is grey, that outside it is white. Only the geometry of the exterior region is that of Schwarzschild. The curve separating the grey and white regions is the geodesic world line of the surface of the collapsing star (equations [31.10] or [32.10] with

r_{max} = R_{i} = 10 M)

. This world line is parameterized by proper time,

τ

, as measured by an observer who sits on the surface of the star; the radial light cones, as calculated from

d s^{2} = 0

, are attached to it.
Notice that, although the shapes of the light cones are not all the same relative to Schwarzschild coordinates or relative to Eddington-Finkelstein coordinates, they are all the same relative to KruskalSzekeres coordinates. This is because light rays travel along 45-degree lines in the

u, v

-plane

(d v = \pm d u)

, but they travel along curved paths in the

r, t

-plane and

r, \tilde{V}

-plane.

The Kruskal-Szekeres spacetime diagram shown here is related to the Schwarzschild diagram by equations (31.13) plus a translation of Schwarzschild time:

t ⟶ t + 42.8 M

. The Eddington-Finklestein diagram is related to the Schwarzschild diagram by

\tilde{V} = t + r^{*} = t + r + 2 M \ln | r / 2 M - 1 |

(see Box 31.2).
It is evident from these diagrams that the free-fall collapse is characterized by a constantly diminishing radius, which drops from

R = 10 M

R = 0

in a finite and short comoving proper time interval,

Δ τ = 35.1 M

. The point

R = 0

and the entire region

r = 0

outside the star make up a physical "singularity" at which infinite tidal gravitational forces-according to classical, unquantized general relativity-can and do crush matter to infinite density (see end of

§ 31.2

; also

§ 32.6

The Eddington-Finkelstein diagram depicts a series of photons emitted radially from the surface of the collapsing star, and received by an observer at

r = R_{initial} = 10 M

. The observer eventually receives all photons emitted radially from outside the gravitational radius; all photons emitted after the star passes through its gravitational radius eventually get pulled into the singularity at

r = 0

; and any photon emitted radially at the gravitational radius stays at the gravitational radius forever.

Non-free-fall collapse is similar to the collapse depicted here. When pressure gradients are present, only the detailed shape of the world line of the star's surface changes.

Notice how short is the characteristic

e

-folding time for the decay of luminosity and for the radial redshift:

\begin{aligned} τ_{char} & = 2 M \approx 1 \times 10^{- 5} (M / M_{⊙}) sec \\ (32.7) & = (\binom{light-travel time across a flat-space}{distance equal to the gravitational radius}) . \end{aligned}

Here

M_{⊙}

denotes one solar mass.

EXERCISE

The rest of this chapter is Track 2. No previous Track-2 material is needed as preparation for it, but it is needed as preparation for (1) the Track-2 part of Chapter 33 (black holes), and (2) Chapter 34 (singularities and global methods).

Exercise 32.2. REDSHIFTS DURING COLLAPSE

(a) Let a radio transmitter on the surface of a collapsing spherical star emit monochromatic waves of wavelength

λ_{e}

; and let a distant observer, at the same

θ, ϕ

, as the transmitter, receive the waves. Show that at late times the wavelength received varies as

\begin{matrix} (32.8a) & λ_{rec} / λ_{em} \propto e^{t / 4 M} \end{matrix}

[equation (32.6)], where

t

is proper time as measured by the distant observer.
(b) [Track 2] Use kinetic theory for the outgoing photons (conservation of density in phase space: Liouville's theorem;

$ 22.6

) to show that the energy flux of the radiation received (ergs

/ {cm}^{2} \sec

) varies as

\begin{matrix} (32.8b) & F \propto e^{- t / M} \end{matrix}

E_{e}

, and that these neutrinos flow freely outward (negligible absorption in star). Show that the energy of the neutrinos received by a distant observer decreases at late times as

\begin{matrix} (32.9a) & E_{rec} / E_{e} \propto e^{- t / 4 M} \end{matrix}

(d) Show that the flux of neutrino energy dies out at late times as

\begin{matrix} (32.9b) & F \propto e^{- t / 2 M} \end{matrix}

(e) Explain in elementary terms why the decay laws (32.8a) and (32.9a) for energy are the same, but the decay laws (32.8b) and (32.9b) for energy flux are different.
(f) Let a collapsing star emit photons from its surface at the black-body rate

\frac{d N}{d τ} = (1.5 \times 10^{11} \frac{photons}{{cm}^{2} \sec K^{3}}) \times (\binom{surface area}{of star}) \times {(\binom{temperature}{of surface})}^{3}

Let a distant observer count the photons as they pass through his sphere of radius

r ≫ M

. Let him begin his count (time

t = 0

) when he sees (via photons traveling radially outward) the center of the star's surface pass through the radius

r = 3 M

. Show that, in order of magnitude, the time he and his associates must wait, until the last photon that will ever get out has reached them, is

\begin{matrix} (32.9c) & t = (M / M_{⊙}) [8 \times 10^{- 4} + 5 \times 10^{- 5} \log_{10} (T_{11} M / M_{⊙})] seconds, \end{matrix}

where

T_{11}

is the star's surface temperature in units of

10^{11} K

§32.4. COLLAPSE OF A STAR WITH UNIFORM DENSITY AND ZERO PRESSURE

When one turns attention to the interior of a collapsing star and to the precise world line that its surface follows in the Schwarzschild geometry, one encounters rather complicated mathematics. The simplest case to treat is that of a "star" with uniform density and zero pressure; and, indeed, until recently that was the only case which had been treated in detail. The original-and very complete-analysis of the collapse of such a uniform-density "ball of dust" was given in the classic paper of Oppenheimer and Snyder (1939). More recently, other workers have discussed it from slightly different points of view and using different coordinate systems. The approach taken here was devised by Beckedorff and Misner (1962).

Because no pressure gradients are present to deflect their motion, the particles on the surface of any ball of dust must move along radial geodesics in the exterior Schwarzschild geometry. For a ball that begins at rest with finite radius,

R = R_{i}

, at time

t = 0

, the subsequent geodesic motion of its surface is given by equations (31.10):

\begin{matrix} (32.10a) & R = (R_{i} / 2) (1 + \cos η), \\ t = 2 M \ln | \frac{{(R_{i} / 2 M - 1)}^{1 / 2} + \tan (η / 2)}{{(R_{i} / 2 M - 1)}^{1 / 2} - \tan (η / 2)} | \\ (32.10b) & + 2 M {(R_{i} / 2 M - 1)}^{1 / 2} [η + (R_{i} / 4 M) (η + \sin η)] . \end{matrix}

Here

R

is the Schwarzschild radial coordinate (i.e.,

4 π R^{2}

is the star's surface area) at Schwarzschild time

t

. This world line is plotted in Figure 32.1 for

R_{i} = 10 M

, in terms of Schwarzschild coordinates, Kruskal-Szekeres coordinates, and EddingtonFinkelstein coordinates. The proper time read by a clock on the surface of the collapsing star is given by equation (31.10b):

\begin{matrix} (32.10c) & τ = {(R_{i}^{3} / 8 M)}^{1 / 2} (η + \sin η) \end{matrix}

Note that the collapse begins when the parameter

η

is zero (

R = R_{i}, t = τ = 0

); and it terminates at the singularity (

R = 0, η = π

) after a lapse of proper time, as measured on any test particle falling with the dust, equal to

Δ τ = π {(R_{i}^{3} / 8 M)}^{1 / 2}

It is interesting, though coincidental, that this is precisely the time-lapse required for free-fall collapse to infinite density in Newtonian theory [see equation (25.27'), Figure 25.3, and associated discussion].
What is the behavior of the interior of the ball of dust as it collapses? A variety of different interiors for pressureless dust can be conceived (exercise 32.8). But here attention focuses on the simplest of them: an interior that is homogeneous and isotropic everywhere, except at the surface-i.e., an interior locally identical to a dust-filled Friedmann cosmological model (Box 27.1). Is the Friedmann interior to be "open"

(k = - 1)

, "flat"

(k = 0)

, or "closed"

(k = + 1)

? Only the closed case

The collapse, from rest, of a uniform-density ball of "dust":
(1) world line of ball's surface in exterior Schwarzschild coordinates
(2) interior of ball is identical to a portion of a closed Friedmann universe
is appropriate, since one has already demanded [equation (32.10)] that the star be at rest initially (initial rate of change of density equals zero; "moment of maximum expansion").

Using comoving hyperspherical coordinates,

χ, θ, ϕ

, for the star's interior, and putting the origin of coordinates at the star's center, one can write the line element in the interior in the familiar Friedmann form

\begin{matrix} (32.11) & d s^{2} = - d τ^{2} + a^{2} (τ) [d χ^{2} + \sin^{2} χ (d θ^{2} + \sin^{2} θ d ϕ^{2})] \end{matrix}

Here

a (τ)

is given by the familiar cycloidal relation,

\begin{matrix} a = \frac{1}{2} a_{m} (1 + \cos η), \\ (32.12) & τ = \frac{1}{2} a_{m} (η + \sin η); \end{matrix}

and the density is given by

\begin{matrix} (32.13) & ρ = (3 a_{m} / 8 π) a^{- 3} = (3 / 8 π a_{m}^{2}) {[\frac{1}{2} (1 + \cos η)]}^{- 3} \end{matrix}

[see equations (1), (9), (4), and (5) of Box 27.1, with

η

replaced by

η + π

].
There is one possible difficulty with this interior solution. In the cosmological case, the solution was homogeneous and isotropic everywhere. Here homogeneity and isotropy are broken at the star's surface-which lies at some radius

\begin{matrix} (32.14) & χ = χ_{0} \end{matrix}

for all

τ

, as measured in terms of the hyperspherical polar angle

χ

, a comoving coordinate (first picture in Box 27.2). At that surface (i.e., three-dimensional world tube enclosing the star's fluid) the interior Friedmann geometry must match smoothly onto the exterior Schwarzschild geometry. If the match cannot be achieved, then the Friedmann line element (32.11) cannot represent the interior of a collapsing star. An example of a case in which the matching could not be achieved is an interior of uniform and nonzero pressure, as well as uniform density. In that case there would be an infinite pressure gradient at the star's surface, which would blow off the outer layers of the star, and would send a rarefaction wave propagating inward toward its center. The uniform distribution of density and pressure would quickly be destroyed.
For the case of zero pressure, the match is possible. As a partial verification of the match, one can examine the separate and independent predictions made by the interior and exterior solutions for the star's circumference,

C = 2 π R

, as a function of proper time

τ

at the star's surface. The external Schwarzschild solution predicts the cycloidal relation,

\begin{matrix} C = 2 π R = 2 π (R_{i} / 2) (1 + \cos η), \\ (32.15) & τ = {(R_{i}^{3} / 8 M)}^{1 / 2} (η + \sin η) \end{matrix}

[equations (32.10)]. The interior Friedmann solution predicts a similar cycloidal relation:

\begin{matrix} C = 2 π R = 2 π a \sin χ_{0} = 2 π (\frac{1}{2} a_{m} \sin χ_{0}) (1 + \cos η), \\ (32.16) & τ = \frac{1}{2} a_{m} (η + \sin η) . \end{matrix}

The two predictions agree perfectly for all time if and only if

\begin{matrix} (32.17a) & R_{i} = a_{m} \sin χ_{0}, \\ (32.17b) & M = \frac{1}{2} a_{m} \sin^{3} χ_{0} . \end{matrix}

A more complete verification of the match is given in exercise 32.4.
For further insight into this idealized model of gravitational collapse, see Box 32.1.

Exercise 32.3. EMBEDDING DIAGRAMS AND PHOTON PROPAGATION FOR COLLAPSING STAR

Verify in detail the features of homogeneous collapse described in Box 32.1.

Exercise 32.4. MATCH OF FRIEDMANN INTERIOR TO SCHWARZSCHILD EXTERIOR

The Einstein field equations are satisfied on a star's surface if and only if the intrinsic and extrinsic geometries of the surface's three-dimensional world tube are the same, whether measured on its interior or on its exterior (see

§ 21.13

for proof and discussion). Verify that for the collapsing star discussed above, the intrinsic and extrinsic geometries match at the join between the Friedmann interior and the Schwarzschild exterior. [Hints: (a) Use

η, θ, ϕ

, as coordinates on the world tube of the star's surface, and show that the intrinsic geometry has the same line element

\begin{matrix} (32.18a) & d s^{2} = a^{2} (η) [- d η^{2} + \sin^{2} χ_{0} (d θ^{2} + \sin^{2} θ d ϕ^{2})], \end{matrix}

whether measured in the Schwarzschild exterior or in the Friedmann interior. (b) Show that the extrinsic curvature of the world tube has the same components

\begin{matrix} (32.18b) & K_{η η} = K_{η θ} = K_{η ϕ} = K_{θ ϕ} = 0, \\ K_{θ θ} = K_{ϕ ϕ} / \sin^{2} θ = - a (η) \sin χ_{0} \cos χ_{0}, \end{matrix}

whether measured in the Schwarzschild exterior or in the Friedmann interior.]

Exercise 32.5. STARS THAT COLLAPSE FROM INFINITY

(a) Patch together a truncated Schwarzschild geometry and the geometry of a truncated "flat" (

k = 0

), dust-filled Friedmann universe to obtain a model of a star that collapses from rest at an infinite initial radius. [Hint: The world line of the star's surface in the Schwarzschild geometry is given by equations (31.2).]
(b) Similarly patch together a truncated Schwarzschild geometry and the geometry of a truncated "open"

(k = - 1)

, dust-filled Friedmann universe to obtain a star which collapses from infinity with finite initial inward velocity.

Box 32.1 AN IDEALIZED COLLAPSING STAR WITH FRIEDMANN INTERIOR AND SCHWARZSCHILD EXTERIOR

(See §32.4 and exercises 32.3 and 32.4 for justification of the results described here.)

Initial State

(1) Take a Friedmann universe of radius

a = a_{m}

at its moment of maximum expansion,

η = 0

; and slice off and discard the region

χ_{0} < χ \leq π

, where

χ_{0}

is some angle less than

π / 2

. (2) Take a Schwarzschild geometry of mass

M = (a_{m} / 2) \sin^{3} χ_{0}

at the moment

t = 0

; and slice off and discard the region

r < R_{i} = a_{m} \sin χ_{0}

. (3) Glue the remaining pieces of Friedmann and Schwarzschild geometry together smoothly along their cut surfaces. The resultant object will be a momentarily static star of uniform density

ρ_{i} = 3 / (8 π a_{m}^{2})

, of mass

M = (a_{m} / 2) \sin^{3} χ_{0}

, and of radius

R_{i} =

a_{m} \sin χ_{0}

Subsequent Evolution

Release this star from its intial state, and let it collapse in accord with Einstein's field equations. The interior, truncated Friedmann universe and the exterior, truncated Schwarzschild geometry will evolve just as though they had never been cut up and patched together; and this evolution will preserve the smoothness of the match between interior and exterior!

Details of the Collapse

Probe the details of the collapse using sequences of embedding diagrams (histories

A B C D

and

A W X Y

), and using photons that propagate radially outward (photons

α, β, γ, δ, ϵ)

. The example shown here has

χ_{0} = 0.96

and

R_{i} / M = 2 / \sin^{2} χ_{0} = 3

History of Collapse as Probed by Hypersurfaces ABCD:

(1) Initial configuration,

A - A^{'} - A^{''}

, is that constructed by cutting and sewing at times

η = t = 0

.
(2) Each subsequent configuration has as its interior a slice of constant Friedmann time

η

.
(3) The interior remains always a spherical cup of half-angle

χ_{0}

; but it contracts from radius

R = R_{i} = a_{m} \sin χ_{0}

R = 0

as time increases.
(4) The matter in the star is all crushed simultaneously to infinite density when

R

reaches zero, and the external Schwarzschild "funnel" develops a cusp-like singularity at that point.
(5) As time increases further, this cusp pulls the region

r < 2 M

of the funnel into

r = 0

so fast that the outward-traveling photon

δ

is gobbled up and crushed.

These embedding diagrams must be rotated about the vertical axes in order to become 2-dimensional surfaces analogous to Figure 23.1.

Box 32.1 (continued)

History as Probed by Hypersurfaces AWXY
(1) Initial configuration,

A - A^{'} - A^{''}

, is again that constructed by cutting and sewing at

η = t = 0

.
(2) Subsequent hypersurfaces are very different from

η =

const.
(3) As time passes, a neck develops in the geometry just outside the surface of the star.
(4) This neck becomes tighter and tighter and then pinches off, leaving the star completely isolated from the rest of the universe, and leaving a deadly cusp-like singularity in the exterior geometry where the star used to be.
(5) The isolated star, in its own little closed universe, continues to contract until it is crushed to infinite density, while the external geometry begins to develop another neck and the cusp quickly gobbles up photon

δ

The extreme difference between histories

A B C D

and

A W X Y

dramatizes the "many-fingered time" of general relativity. The hypersurface on which one explores the geometry can be pushed ahead faster in time in one region, at the option of the party of explorers. Thus whether one region of the star collapses first, or another, or the entire star collapses simultaneously, is a function both of the spacetime geometry and of the choice of slicing. The party of explorers has this choice of slicing at their own control, and thus they themselves to this extent govern what kind of spacelike slices they will see as their exploration moves forward in time. The spacetime geometry that they slice, however, is in no way theirs to control or to change. To the extent that their masses are negligible and they serve merely as test objects, they have no influence whatsoever on the spacetime. It was fixed completely by the specification of the initial conditions for the collapse. In brief, spacetime is four-dimensional and slices are only three-dimensional (and in the pictures here look only two-dimensional or one-dimensional). Any one set of slices captures only a one-sided view of the whole story. To see the entire picture one must either examine the dynamics of the geometry as it reveals itself in varied choices of the slicing or become accustomed to visualizing the spacetime geometry as a whole.

§32.5. SPHERICALLY SYMMETRIC COLLAPSE WITH INTERNAL PRESSURE FORCES

So far as the external gravitational field is concerned, the only difference between a freely collapsing star and a collapsing, spherically symmetric star with internal pressure is this: that the surfaces of the two stars move along different world lines in the exterior Schwarzschild geometry. Because the exterior geometry is the same in both cases, the qualitative aspects of free-fall collapse as described in the last section can be carried over directly to the case of nonnegligible internal pressure.

An important and fascinating question to ask is this: can large internal pressures in any way prevent a collapsing star from being crushed to infinite density by infinite tidal gravitational forces? From the Kruskal-Szekeres diagram of Figure 32.1,b, it is evident that, once a star has passed inside its gravitational radius (

R < 2 M

), no internal pressures, regardless of how large they may be, can prevent the star's surface from being crushed in a singularity. The surface must move along a time-like world line, and all such world lines inside

r = 2 M

hit

r = 0

. Although there is no such theorem now available, one can reasonably conjecture that, if the surface of a spherical configuration is crushed in the

r = 0

singularity, the entire interior must also be crushed.

The details of the interior dynamics of a spherically symmetric collapsing star with pressure are not so well-understood as the exterior Schwarzschild dynamics. However, major advances in one's understanding of the interior dynamics are now being made by means of numerical computations and analytic analyses [see Misner (1969a) for a review]. In these computations and analyses, no new features (at least, no unexpected ones) have been encountered beyond those that occurred in the simple uniform-density, free-fall collapse of the last section.

Exercise 32.6. GENERAL SPHERICAL COLLAPSE: METRIC IN COMOVING COORDINATES

Consider an inhomogeneous star with pressure, undergoing spherical collapse. Spherical symmetry alone is enough to guarantee the existence of a Schwarzschild coordinate system

(t, r, θ, ϕ)

throughout the interior and exterior of the star [see equation (32.2) and preceding discussion]. Label each spherical shell of the star by a parameter

a

, which tells how many baryons are contained interior to that shell. Then

r (a, t)

is the world line of the shell with label

a

. The expression for these world lines can be inverted to obtain

a (t, r)

, the number of baryons interior to radius

r

at time

t

. Show that there exists a new time coordinate

\tilde{t} (t, r)

, such that the line element (32.2), rewritten in the coordinates

(\tilde{t}, a, θ, ϕ)

, has the form

\begin{aligned} (32.19a) & d s^{2} = - e^{2 \tilde{Φ}} d {\tilde{t}}^{2} + {[\frac{(\partial r / \partial a)_{\tilde{t}} d a}{Γ}]}^{2} + r^{2} d Ω^{2}, \\ (32.19b) & \tilde{Φ} = \tilde{Φ} (\tilde{t}, a), r = r (\tilde{t}, a), Γ = Γ (\tilde{t}, a) . \end{aligned}

Spherical collapse with pressure is qualitatively the same as without pressure

These are "comoving, synchronous coordinates" for the stellar interior.

Exercise 32.7. ADIABATIC SPHERICAL COLLAPSE: EQUATIONS OF EVOLUTION [Misner and Sharp (1964)]

Describe the interior of a collapsing star by the comoving, synchronous metric (32.19), by the number density of baryons

n

, by the total density of mass-energy

ρ

, and by the pressure

p

. The 4 -velocity of the star's fluid is

\begin{matrix} (32.20) & u = e^{- \tilde{ϕ}} \partial / \partial \tilde{t} \end{matrix}

since the fluid is at rest in the coordinate system. Let a dot denote a proper time derivative as seen by the fluid-e.g.,

\dot{n} \equiv u [n] = e^{- \tilde{Φ}} (\partial n / \partial \tilde{t})_{a}

and let a prime denote a partial derivative with respect to baryon number,-e.g.

n^{'} \equiv (\partial n / \partial a)_{\tilde{t}} .

Denote by

U

the rate of change of

(1 / 2 π) \times

(circumference of shell), as measured by a man riding in a given shell:

\begin{matrix} (32.21a) & U \equiv \dot{r} \end{matrix}

and denote by

m (\tilde{t}, a)

the "total mass-energy interior to shell

a

at time

\tilde{t}

\begin{matrix} (32.2lb) & m (\tilde{t}, a) \equiv \int_{0}^{a} 4 π r^{2} ρ (\tilde{t}, a) r^{'} d a \end{matrix}

(See Box 23.1 for discussion of this method of localizing mass-energy.) Assume that the collapse is adiabatic (no energy flow between adjacent shells; stress-energy tensor entirely that of a perfect fluid).
(a) Show that the equations of collapse [baryon conservation, (22.3); local energy conservation, (22.11a); Euler equation, (22.13); and Einstein field equations (ex. 14.16)] can be reduced to the following eight equations for the eight functions

\tilde{Φ}, Γ, r, n, ρ, p, U, m

\begin{matrix} (32.22h) & \begin{aligned} \dot{r} & = U & (dynamic equation for r); \\ \frac{{(n r^{2})}^{\cdot}}{n r^{2}} & = - \frac{U^{'}}{r^{'}} & (dynamic equation for n); \\ \frac{\dot{ρ}}{ρ + p} & = \frac{\dot{n}}{n}, & \begin{matrix} except at a shock front, where adiabaticity \\ breaks down \end{matrix} \\ \dot{U} & = - \frac{Γ^{2}}{ρ + p} \frac{p^{'}}{r^{'}} - \frac{m + 4 π r^{3} p}{r^{2}} & (dynamic equation for ρ); \\ p & = p (n, ρ) & (equation of state); \\ {\tilde{Φ}}^{'} & = - p^{'} / (ρ + p), \tilde{Φ} = 0 at star's surface & (source equation for \tilde{Φ}); \\ m^{'} & = 4 π r^{2} ρ r^{'}, & m = 0 at a = 0, \\ Γ & (source equation for m); \\ Γ & sign (r^{'}) {(1 + U^{2} - 2 m / r)}^{1 / 2} & (algebraic equation for Γ). \end{aligned} \end{matrix}

(b) The preceding equations are in a form useful for numerical calculations. [For particular numerical solutions and for the handling of shocks, see May and White (1966).] For analytic work it is often useful to replace (32.22b) by

\begin{matrix} (') & n = Γ / (4 π r^{2} r^{'}) \end{matrix}

and ( 32.22 d ) by

\begin{matrix} (') & \dot{m} = - 4 π r^{2} p U \end{matrix}

Derive these equations.
(c) Explain why equations

(32.22 g)

and

(32.22 d^{'})

justify the remarks made in Box 23.1 about localizability of energy.

Exercise 32.8. ANALYTIC SOLUTIONS FOR PRESSURE-FREE COLLAPSE [Tolman (1934b); Datt (1938)]

Show that the general solution to equations (32.22) in the case of zero pressure can be generated as follows.
(a) Specify the mass inside shell

a, m (a)

; by equation (

32.22 d^{'}

), with

p = 0

, it will not change with time

\tilde{t}

.
(b) Assume that all the dust particles have rest masses

μ

that depend upon radius,

μ (a)

; so

\begin{matrix} (32.23a) & ρ = μ n \end{matrix}

Γ

from the equation

\begin{matrix} (32.23b) & Γ = m^{'} / μ \end{matrix}

it will be independent of

\tilde{t}

.
(d) Specify an initial distribution of circumference

2 π r

as function of

a

, and solve the dynamic equation

\begin{matrix} (32.23c) & {(\frac{\partial r}{\partial t})}^{2} - \frac{2 m (a)}{r} = Γ^{2} (a) - 1 \end{matrix}

to obtain the subsequent evolution of

r (\tilde{t}, a)

. Notice that this equation has identically the same form as in Newtonian theory!
(e) Calculate the remaining quantities of interest from the algebraic equations

\begin{matrix} (32.23~d) & d s^{2} = - \tilde{d t^{2}} + {(r^{'} d a / Γ)}^{2} + r^{2} d Ω^{2} \\ (32.23e) & ρ = μ n = m^{'} / (4 π r^{2} r^{'}) \\ (32.23f) & \tilde{Φ} = 0, U = \partial r / \partial \tilde{t} \end{matrix}

[Note: In this solution, successive "shells" may pass through each other, producing a surface of infinite density as they do

(r^{'} ⟶ 0

where

m^{'} \neq 0)

, since there is no pressure built up to stop shell crossing. When this happens, the coordinate system becomes pathological (

a

no longer increases monotonically outward), but spacetime remains well-behaved. The surface of infinite density (1) produces negligible tidal forces on neighboring dust particles; and (2) like the surface layers of

$ 21.13

, it is an idealization that gets smeared down to finite density by finite pressure.]

Exercise 32.9. COLLAPSE WITH UNIFORM DENSITY

Recover the Friedmann-Schwarzschild solution for collapse with uniform density and zero pressure by specifying appropriate forms for

m (a)

and

r (a)

in the prescription of exercise 32.8. In the interior of the star, give the dust particles nonzero rest masses,

μ =

constant

\neq 0

; in the exterior give them zero rest masses,

μ = 0

("imaginary dust particles" in vacuum). Reduce the resulting metric ( 32.23 d ) to that of Friedmann inside the star, and to that of Novikov for the Schwarzschild geometry outside the star [equations (31.12)].

§32.6. THE FATE OF A MAN WHO FALLS INTO THE SINGULARITY AT $r = 0$

The effect of tidal forces on the body of a man falling into the

r = 0

singularity:

Stage 1: body resists deformation; stresses build up

Consider the plight of an experimental astrophysicist who stands on the surface of a freely falling star as it collapses to

R = 0

.
As the collapse proceeds toward

R = 0

, the various parts of the astrophysicist's body experience different gravitational forces. His feet, which are on the surface of the star, are attracted toward the star's center by an infinitely mounting gravitational force; while his head, which is farther away, is accelerated downward by a somewhat smaller, though ever rising force. The difference between the two accelerations (tidal force) mounts higher and higher as the collapse proceeds, finally becoming infinite as

R

reaches zero. The astrophysicist's body, which cannot withstand such extreme forces, suffers unlimited stretching between head and foot as

R

drops to zero.
But this is not all. Simultaneous with this head-to-foot stretching, the astrophysicist is pulled by the gravitational field into regions of spacetime with ever-decreasing circumferential area,

4 π r^{2}

. In order to accomplish this, tidal gravitational forces must compress the astrophysicist on all sides as they stretch him from head to foot. The circumferential compression is actually more extreme than the longitudinal stretching; so the astrophysicist, in the limit

R ⟶ 0

, is crushed to zero volume and indefinitely extended length.
The above discussion can be put on a mathematical footing as follows.
There are three stages in the killing of the astrophysicist: (1) the early stage, when his body successfully resists the tidal forces; (2) the intermediate stage, when it is gradually succumbing; and (3) the final stage, when it has been completely overwhelmed.
During the early stage, one can analyze the tidal forces by means of the equation of geodesic deviation, evaluated in the astrophysicist's orthonormal frame

ω^{\hat{γ}}, ω^{\hat{ρ}}

ω^{\hat{θ}}, ω^{\hat{ϕ}}

(see

§ 31.2

). In this frame, the nonvanishing components of the Riemann curvature tensor are given by equations (31.6):

\begin{matrix} (32.24a) & \begin{array}{ll} R_{\hat{γ} \hat{ρ} \hat{r} \hat{ρ}} = - 2 M / r^{3}, & R_{\hat{γ} \hat{θ} \hat{τ} \hat{θ}} = R_{\hat{τ} \hat{ϕ} \hat{γ} \hat{ϕ}} = M / r^{3}, \\ R_{\hat{θ} \hat{ϕ} \hat{θ} \hat{ϕ}} = 2 M / r^{3}, & R_{\hat{ρ} \hat{ρ} \hat{ρ} \hat{θ}} = R_{\hat{ρ} \hat{ϕ} \hat{ϕ} \hat{ϕ} \hat{h}} = - M / r^{3} . \end{array} \end{matrix}

The equation of geodesic deviation says that two freely moving particles, momentarily at rest in the astrophysicist's local inertial frame, and separated by the 3-vector

ξ = ξ^{\hat{c}} e_{\hat{j}}

must accelerate apart with a relative acceleration given by

\begin{aligned} D^{2} ξ^{\hat{j}} / d τ^{2} & = - R_{\hat{γ} \hat{k} \hat{r}} ξ^{\hat{k}} = - R_{\hat{j} \hat{τ} \hat{τ} \hat{ξ}} ξ^{\hat{k}} \\ = - R_{\hat{τ} \hat{j} \hat{τ} \hat{k}} ξ^{\hat{k}} . \end{aligned}

Using the components (32.24a) of the curvature tensor, one sees that

\begin{aligned} D^{2} ξ^{\hat{ρ}} / d τ^{2} = + (2 M / r^{3}) ξ^{\hat{ρ}} \\ (32.24b) & D^{2} ξ^{\hat{θ}} / d τ^{2} = - (M / r^{3}) ξ^{\hat{θ}} \\ D^{2} ξ^{\hat{ρ}} / d τ^{2} = - (M / r^{3}) ξ^{\hat{ϕ}} \end{aligned}

To apply these equations to the astrophysicist's body, idealize it (for simplicity) as a homogeneous rectangular box of mass

μ \approx 165

pounds

\approx 75 kg

, of length

ℓ \approx 70

inches

\approx 1.8 m

in the

e_{\hat{ρ}}

direction, and of width and depth

w \approx 10

inches

\approx 0.2 m

in the

e_{\hat{θ}}

and

e_{\hat{ϕ}}

directions. Then calculate the stresses that must be set up in this idealized body to prevent its particles from moving along diverging (and converging) geodesics.
From the form of equations (32.24), it is evident that the principal directions of the stress will be

e_{\hat{ρ}}, e_{\hat{θ}}

, and

e_{\hat{ϕ}}

(i.e., in the

e_{\hat{ρ}}, e_{\hat{θ}}, e_{\hat{ϕ}}

basis, the stress tensor will be diagonal). The longitudinal component of the stress, at the astrophysicist's center of mass, can be evaluated as follows. A volume element of his body with mass

d μ

, located at a height

h

above the center of mass (distance

h

measured along

e_{\hat{ρ}}

direction) would accelerate with

a = (2 M / r^{3}) h

away from the center of mass, if it were allowed to move freely. To prevent this acceleration, the astrophysicist's muscles must exert a force

d F = a d μ = (2 M / r^{3}) h d μ

This force contributes to the stress across the horizontal plane (

e_{\hat{θ}} \land e_{\hat{ϕ}}

plane) through the center of mass. The total force across that plane is the sum of the forces on all mass elements above it (which is also equal to the sum of the forces on the mass elements below it):

\begin{aligned} F & = \int_{(region above plane)} a d μ = \int_{0}^{1 / 2} (\frac{2 M h}{r^{3}}) (\frac{μ}{ℓ w^{2}}) (w^{2} d h) \\ = \frac{1}{4} \frac{μ M ℓ}{r^{3}} . \end{aligned}

The stress is this force divided by the cross-sectional area

w^{2}

, with a minus sign because it is a tension rather than a pressure:

\begin{matrix} (32.25a) & T_{\hat{ρ} \hat{ρ}} = - \frac{1}{4} \frac{μ M ℓ}{w^{2} r^{3}} \approx - 1.1 \times 10^{15} \frac{M / M_{⊙}}{(r / 1 km)^{3}} \frac{dynes}{{cm}^{2}} \end{matrix}

The components of the stress in the

e_{\hat{θ}}

and

e_{\hat{ϕ}}

directions at the center of mass are, similarly,

\begin{matrix} (32.25b) & T_{\hat{θ} \hat{θ}} = T_{\hat{ϕ} \hat{ϕ}} = + \frac{1}{8} \frac{μ M}{l r^{3}} \approx + 0.7 \times 10^{13} \frac{M / M_{⊙}}{(r / 1 km)^{3}} \frac{dynes}{{cm}^{2}} . \end{matrix}

(Recall that one atmosphere of pressure is

1.01 \times 10^{6}

dynes

/ {cm}^{2}

Stage 2: body gives way; man dies

Stage 3: body gets crushed and distended

The human body cannot withstand a tension or pressure of

\geq 100

atmospheres

\approx 10^{8}

dynes

/ {cm}^{2}

without breaking. Consequently, an astrophysicist on a freely collapsing star of one solar mass will be killed by tidal forces when the star's radius is

R \sim 200 km ≫ 2 M \approx 3 km

By the time the star is much smaller than its gravitational radius, the baryons of the astrophysicist's body are moving along geodesics; his muscles and bones have completely given way. In this final stage of collapse, the timelike geodesics are curves along which the Schwarzschild "time"-coordinate,

t

, is almost constant [

c f

. the narrowing down of the light cones near

r = 0

in Figure 32.1, a; also equation (31.2) in the limit

r ≪ 2 M]

. The astrophysicist's feet touch the star's surface at one value of

t

-say

t = t_{f}

-while his head moves along the curve

t = t_{h} > t_{f}

. Consequently, the length of the astrophysicist's body increases according to the formula

\begin{aligned} ℓ_{astroph} & = {[g_{t t} (R)]}^{1 / 2} [t_{h} - t_{f}] = [2 M / R]^{1 / 2} [t_{h} - t_{f}] \\ (32.26a) & \propto R^{- 1 / 2} \propto {(τ_{collapse} - τ)}^{- 1 / 3} \end{aligned}

Here

τ = [- \int^{R} {| g_{r r} |}^{1 / 2} d r +

constant

]

is proper time as it would be measured by the astrophysicist if he were still alive, and

τ_{collapse}

is the time at which he hits

r = 0

. The gravitational field also constrains the baryons of the astrophysicist's body to fall along world lines of constant

θ

and

ϕ

during the final stages of collapse. Consequently, his cross-sectional area decreases according to the law

\begin{matrix} (32.26b) & a_{astroph} = {[g_{θ θ} (R) g_{ϕ ϕ} (R)]}^{1 / 2} Δ θ Δ ϕ \propto R^{2} \propto {(τ_{collapse} - τ)}^{4 / 3} \end{matrix}

By combining equations (

32.26 a, b

), one sees that the volume of the astrophysicist's body decreases, during the last few moments of collapse, according to the law

\begin{matrix} (32.26c) & V_{astroph} = ℓ_{astroph} Q_{astroph} \propto R^{3 / 2} \propto (τ_{collapse} - τ) \end{matrix}

This crushing of matter to infinite density by infinitely large tidal gravitational forces can occur not only on the surface of the collapsing star, but also at any other point along the

r = 0

singularity outside the surface of the star. Hence, any foolish rocketeer who ventures below the radius

r = 2 M

of the external gravitational field is doomed to destruction.

For further discussion of spacetime singularities, and of the possibility that quantum gravitational effects might force a reconsideration of the singularities predicted by classical gravitation theory, see Chapter 30,

§ 34.6

, and Chapter 44.

§32.7. REALISTIC GRAVITATIONAL COLLAPSEAN OVERVIEW

Review of spherical collapse

Instability, implosion, horizon, and singularity; these are the key stages in the spherical collapse of any star. Instability: The star, having exhausted its nuclear fuel, and having contracted slowly inward, begins to squeeze its pressure-sustaining electrons or photons onto its atomic nuclei; this softens the equation of state, which induces an instability [see, e.g.,

§ § 10.15

and 11.4 of Zel'dovich and Novikov (1971)
for details]. Implosion: Within a fraction of a second the instability develops into a full-scale implosion; for realistic density distributions, the stellar core falls rapidly inward on itself, and the outer envelopes trail along behind [see, e.g., the numerical calculations of Colgate and White (1966), Arnett (1966, 1967), May and White (1966), and Ivanova, Imshennik, and Nadezhin (1969)]. Horizon: In the idealized spherical case, the star's surface falls through its gravitational radius ("horizon"; end of communication with the exterior; point of no return). From the star's vantage point this happens after a finite, short lapse of proper time. But from an external vantage point the star requires infinite time to reach the horizon, though it becomes black exponentially rapidly in the process [e-folding time

\sim M \sim 10^{- 5} (M / M_{⊙})

sec]. The result is a "black hole", whose boundary is the horizon (gravitational radius), and whose interior can never communicate with the exterior. Singularity: From the star's interior vantage point, within a short proper time interval

Δ τ \sim M \sim 10^{- 5} (M / M_{⊙})

sec after passing through the horizon, a singularity is reached (zero radius, infinite density, infinite tidal gravitational forces).

Does this basic picture-instability, implosion, horizon, singularity-have any relevance for real stars? Might complications such as rotation, nonsphericity, magnetic fields, and neutrino fluxes alter the qualitative picture? No, not for small initial perturbations from sphericity. Perturbation theory analyses described in Box 32.2 and exercise 32.10 show that realistic, almost-spherically symmetric collapse, like idealized collapse, is characterized by instability, implosion, horizon; and Penrose (1965b; see

$ 34.6

) proves that some type of singularity then follows.

Highly nonspherical collapse is more poorly understood, of course. Nevertheless, a number of detailed calculations and precise theorems point with some confidence to two conclusions: (1) horizons (probably) form when and only when a mass

M

gets compacted into a region whose circumference in EVERY direction is

C \leq 4 π M

(Box

32.3)

; (2) the external gravitational field of a horizon (black hole), after all the "dust" and gravitational waves have cleared away, is almost certainly the Kerr-Newman generalization of the Schwarzschild geometry (Chapter 33). If so, then the external field is determined uniquely by the mass, charge, and angular momentum that went "down the hole." (This nearly proved theorem carries the colloquial title "A black hole has no hair.")

The interior of the horizon, and the endpoint (if any) of the collapse are very poorly understood today. The various possibilities will be reviewed in Chapter 34. That a singularity occurs one can state with much certainty, thanks to theorems of Penrose, Hawking, and Geroch. But whether all, only some, or none of the collapsing matter and fields ultimately encounter the singularity one does not know.

EXERCISES

Summary of 1972
knowledge about realistic, nonspherical collapse:
(1) horizon
(2) black hole
(3) singularity

Exercise 32.10. PRICE'S THEOREM FOR A SCALAR FIELD
Exercise 32.10. PRICE'S THEOREM FOR A SCALAR FIELD [See Price (1971, 1972a), also Thorne (1972), for more details than are presented here.]

A collapsing spherical star, with an arbitrary nonspherical "scalar charge distribution," generates an external scalar field

Φ

. The vacuum field equation for

Φ

◻ Φ = Φ_{; α}^{α} = 0

. Ignore the back-reaction of the field's stress-energy on the geometry of spacetime.

Box 32.2 COLLAPSE WITH SMALL NON-SPHERICAL PERTURBATIONS [based on detailed calculations by Richard H. Price (1971, 1972a,b)].

A. Density Perturbations

When star begins to collapse, it possesses a small nonspherical "lump" in its density distribution.
As collapse proceeds, lump grows larger and larger [instability of collapse against small per-turbations-a phenomenon well known in Newtonian theory; see, e.g., Hunter (1967); Lin, Mestel, and Shu (1965)].
The growing lump radiates gravitational waves.
Waves of short wavelength $(λ ≪ M)$ , emitted from near horizon $(r - 2 M ≦ M)$ , partly propagate to infinity and partly get backscattered by the "background" Schwarzschild curvature of spacetime. Backscattered waves propagate into horizon (surface of black hole; gravitational radius) formed by collapsing star.
Waves of long wavelength $(λ ≫ M)$ , emitted from near horizon $(r - 2 M ≦ M)$ , get fully backscattered by spacetime curvature; they never reach out beyond $r \sim 3 M$ ; they end up propagating "down the hole."
Is lump on star still there as star plunges through horizon, and does star thereby create a deformed (lumpy) horizon? Yes, according to calculations.
But external observers can only learn about existence of "final lump" by examining deformation (quadrupole moment) in final gravitational field. That final deformation in field does not and cannot propagate outward with infinite speed (no instantaneous "action at a distance"). It propagates with speed of light, in form of gravitational waves with near-infinite wavelength (infinite redshift from edge of horizon to any external radius). Deformation in final field, like any other wave of long wavelength, gets fully backscattered by curvature of spacetime at $r ≦ 3 M$ ; it cannot reach external observers. External observers can never learn of existence

of final lump. Final external field is perfectly spherical, lump-free, Schwarzschild geometry!
Even in region of backscatter $(2 M < r ≦ 3 M)$ , final external field is lump-free. Backscattered waves, carrying information about existence of final lump, interfere destructively with outgoing waves carrying same information. Result is destruction of all deformation in external field and in horizon!
Final black hole is a Schwarzschild black hole!

B. Perturbations in Angular Momentum

When star begins to collapse, it possesses a small, nonzero intrinsic angular momentum ("spin") $S$ .
As collapse proceeds, $S$ is conserved (except for a tiny, negligible change due to angular momentum carried off by waves; that change is proportional to square of amplitude of waves, i.e., to square of amplitude of perturbations in star, i.e., to $S^{2}$ ).
Consequently, external field always and everywhere carries imprint of angular momentum $S$ (on imprints, see Chapter 19). There is no need for that imprint to propagate outward from near horizon. Moreover, it could not so propagate even if it tried, because of the conservation law for $S$ (absence of dipole gravitational waves; see $$ § 36.1$ and 36.10 ).
Hence, the final external field is that of an undeformed, slowly rotating black hole:

\begin{aligned} d s^{2} = - & \underset{Schwarzschild geometry}{\underset{⏟}{(1 - \frac{2 M}{r}) d t^{2} + \frac{d r^{2}}{1 - 2 M / r} + r^{2} d Ω^{2}}} \\ - \underset{⏟}{\frac{4 S \sin θ}{r^{2}}) (r \sin θ d ϕ) d t .} \end{aligned}

rotational imprint, see exercise 26.1; also Chapter 19.

Here the polar axis has been oriented along

S

C. Perturbations in Electromagnetic Field

Star possesses a magnetic field generated by currents in its interior, and an electric field due to an arbitrary internal charge distribution; and electromagnetic radiation is emitted by its hot matter. For simplicity, $S$ is assumed zero.
Evolution of external electromagnetic field is similar to evolution of perturbations in external gravitational field. Distant observer can never learn "final" values of changeable quantities (magnetic dipole moment, electric dipole moment, quadrupole moments, . . .). Final values try to propagate out from horizon, carried by electromagnetic waves of near-infinite wavelength. But they cannot get out: spacetime curvature reflects them back down the hole; and they superpose destructively with their outgoing counterparts, to produce zero net field.
By contrast with all other quantities, which are changeable, the electric monopole moment (total flux of electric field; equal to $4 π$ times total electric charge) is conserved. It never changes from before star collapses, through the collapse stage, into the quiescent black-hole stage.
Hence, the final external electromagnetic field is a spherically symmetric coulomb field
$E = (Q / r^{2}) e_{r})$ as measured by static
$B = 0$ observer $(r, θ, ϕ$ , constant $)$ ;
and the final spacetime geometry is that of Reissner and Nordstrom (charged black hole; see exercises 31.8 and 32.1 ):

\begin{aligned} d s^{2} = - (1 & - \frac{2 M}{r} + \frac{Q^{2}}{r^{2}}) d t^{2} \\ + \frac{d r^{2}}{(1 - 2 M / r + Q^{2} / r^{2})} + r^{2} d Ω^{2} . \end{aligned}

D. Generalization; Price's Theorem

Let the star generate a "zero-rest-mass, integerspin field." ["Zero rest mass" refers to the quantized particles associated with the classical field. Classically it means the field has a Cou-lomb-law ( $1 / r$ ) fall off at large distances. The spin also is a property of the quantized particles; classically it is most easily visualized as describing the symmetries of a monochromatic plane wave under rotations about the direction of propagation; see $§ 35.6$ . A scalar field has spin zero; an electromagnetic field has spin one; Einstein's gravitational field has spin two; . . . . Of such fields, only gravitational $(s = 2)$ and electromagnetic ( $s = 1$ ) are known to exist in the real universe. See, e.g., Dirac (1936), Gårding (1945), Bargmann and Wigner (1948), Penrose (1965a), for further discussion.]
Let the spin-s field be sufficiently weak that its stress-energy perturbs the star's external, Schwarzschild geometry only very slightly.
Resolve the external field into spherical harmonics (scalar spherical harmonics for $s = 0$ ; vector spherical harmonics for $s = 1$ ; tensor spherical harmonics for $s \geq 2$ ); and label the spherical harmonics by the usual integer $ℓ (ℓ =$ 0 for monopole; $ℓ = 1$ for dipole; $ℓ = 2$ for quadrupole; etc.).
All multipole fields with $ℓ < s$ are conserved during the collapse (theorem from classical radiation theory). A scalar field $(s = 0)$ conserves
nothing. The electromagnetic field $(s = 1)$ conserves only its monopole parts (electric Coulomb field, and vanishing magnetic Coulomb field). The gravitational field $(s = 2)$ conserves its monopole part (with imprint equal to mass), and its dipole parts (with imprints measuring the angular momentum, and the standard gravitational dipole moment-which vanishes if coordinate system is centered on star).
For $ℓ \geq s$ , and only for $ℓ \geq s$ , radiation is possible (scalar waves can have any multipolarity; electromagnetic waves must be dipole and higher; gravitational waves must be quadrupole and higher; see §36.1).
Price's theorem states that, as the nearly spherical star collapses to form a black hole, all things that can be radiated (all multipoles $ℓ \geq s$ ) get radiated completely away-in part "off to infinity"; in part "down the hole" ("what is permitted is compulsory"). The final field is characterized completely by its conserved quantities (multipole moments with $ℓ < s$ ).
For proof of Price's theorem in the case of a scalar field, see exercise 32.10.

E. Generalization to Nonclassical Fields

See Hartle

(1971, 1972)

and Teitelboim (1972b,c) for neutrino fields; Bekenstein (1972a,b) and Teitelboim (1972a) for pion fields.

Box 32.3 COLLAPSE IN ONE AND TWO DIMENSIONS

A. The Question

To produce a black hole (horizon from which nothing can emerge), must one compact matter strongly in all three spatial dimensions, to circumferences

C ≦ 4 π M

(quasispherical compaction); or is it sufficient to compact only in one or two dimensions?

B. The Answer for One Dimension

Consider, as an example readily generalized, the gravitational collapse of a spheroid of dust (zero pressure). Let the spheroid be highly Newtonian (

r ⋙ 2 M

) in its initial, momentary state of rest; and let it be slightly flattened (oblate). In Newtonian theory, any homogeneous, nonrotating spheroid of dust remains homogeneous as it collapses; but its deformations grow [see, e.g., Lin, Mestel, and Shu (1965) for details]. Hence, the spheroid of interest implodes to form a pancake of infinite density but finite mass per unit surface area. The final kinetic energy of the dust particles is roughly equal to their final potential energy:

\begin{aligned} \frac{1}{2} v^{2} \sim \frac{M}{(C / 2 π)} \\ M = mass of spheroid, \\ C = circumference of final pancake. \end{aligned}

Consequently, so long as

C / 2 π ≫ 2 M

, the collapse velocities remain much smaller than light, and the gravitational energy remains much smaller than the rest mass-energy. This means that for

C / 2 π ≫

2 M

, the Newtonian analysis is an excellent approximation to general relativity all the way down to the pancake endpoint. Hence, no horizon can form, hardly any gravitational waves are emitted, and the whole story is exceedingly simple and fully Newtonian. However, since the pancake endpoint is not a singularity of spacetime (see the remarks at end of exercise 32.8), the evolution can proceed beyond it; and as

E

contracts to

≦ 4 π M

, the evolu-
tion will become very complicated and highly relativistic (see the "collapse, pursuit, and plunge scenario" of Figure 24.3).

C. The Answer for Two Dimensions

Consider, as an example not so readily generalized, the gravitational collapse of a homogeneous prolate spheroid of dust, initially highly Newtonian. Such a spheroid collapses to form a thin "thread" or "spindle" [see Lin, Mestel, and Shu (1965)]. Assume that the spheroid is still Newtonian when its threadlike state is reached. It then has a length

ℓ

, a mass per unit length

λ = M / ℓ ≪ 1

, and a rapidly contracting equatorial radius

R ≪ ℓ

. Subsequently, each segment of the thread collapses radially as though it were part of an infinite cylinder. [Ignore the instability of breakup into "beads"; see, e.g., Hunter (1967), Chandrasekhar (1968).] The radial collapse velocity approaches the speed of light and the gravitational energy approaches the rest mass-energy only when the thread has become exceedingly thin,

R ≲ R_{crit} \sim ℓ \exp (- 1 / 4 λ)

. At this stage, relativistic deviations from Newtonian collapse come into play. Thorne (1972) and Morgan and Thorne (1973) have analyzed the relativistic effects using an idealized infinite-cylinder model. The results are very different from either the spherical case or the pancake case. The collapsing cylinder emits a large flux of gravitational waves; but they are powerless to halt the collapse. The collapse proceeds inward to a thread-like singularity, without the creation of any horizon (no black hole!).

D. Objection to the Answer, a Reply, and a Conjecture

One can object that the collapses of both pancake and cylinder can be halted short of their endpoints, especially that of the pancake. As the thickness of

Box 32.3 (continued)

the pancake approaches zero, the vertical pull of gravity remains finite, but the pressure gradient caused by any finite pressure goes to infinity. Hence, pressure halts the collapse. Subsequently the rim of the pancake contracts toward the relativistic regime

C / 2 π \leq 2 M

. In the collapse of a cylinder according to Newtonian theory, with a pressure-density relation

p \propto ρ^{γ}

, the gravitational acceleration

a_{g}

and pressure-buoyancy acceleration

a_{p}

vary as

a_{g} = - 2 λ / R, a_{p} \sim ρ^{- 1} (p / R) \propto ρ^{γ - 1} / R .

Hence, for

γ > 1

(the most common realistic case) pressure halts the collapse, but for

γ < 1

it does
not. Whether this is true also after the relativistic domain is reached, one does not yet know.
Actually, the ability of pressure to halt the collapse is of no importance to the issue of black holes and horizons. The important thing is that in oblate collapse with final circumference

C > 4 π M

, and also in prolate collapse with final thread length

ℓ ≫ 2 M

, no horizons are created. This, coupled with the omnipresent horizons in nearly spherical collapse (Box 32.2) suggests the following conjecture [Thorne (1972)]: Black holes with horizons form when and only when a mass

M

gets compacted into a region whose circumference in EVER Y direction is

C \leq 4 π M

. (Like most conjectures, this one is sufficiently vague to leave room for many different mathematical formulations!)
(a) Resolve the external field into scalar spherical harmonics, using Schwarzschild coordinates for the external Schwarzschild geometry:

\begin{matrix} (32.27a) & Φ = \sum_{i} \frac{1}{r} Ψ_{l} (t, r) Y_{l m} (θ, ϕ) . \end{matrix}

Show that the vacuum field equation reduces to

\begin{matrix} (32.27b) & - Ψ_{1, t t} + Ψ_{ℓ, r^{*} r^{*}} = (1 - \frac{2 M}{r}) (\frac{2 M}{r^{3}} + \frac{ℓ (ℓ + 1)}{r^{2}}) Ψ_{t}, \end{matrix}

where

r^{*}

is the "tortoise coordinate" of

$ 25.5

and Figure 25.4:

\begin{matrix} (32.27c) & r^{*} = r + 2 M \ln (r / 2 M - 1) \end{matrix}

Notice that ( 32.27 b ) is a flat-space, one-dimensional wave equation with effective potential

\begin{matrix} (32.27~d) & V_{eff} (r) = (1 - \frac{2 M}{r}) (\frac{2 M}{r^{3}} + \frac{ℓ (ℓ + 1)}{r^{2}}) . \end{matrix}

Part of this effective potential

[ℓ (ℓ + 1) / r^{2}]

is the "centrifugal barrier," and part

[2 M / r]

is due to the curvature of spacetime. Notice the similarity of this effective potential for scalar waves, to the effective potentials for particles and photons moving in the Schwarzschild geometry,

\begin{matrix} {({\tilde{V}}^{2})}_{particles} = (1 - 2 M / r) (1 + {\tilde{L}}^{2} / r^{2}) \\ {(B^{- 2})}_{photons} = (1 - 2 M / r) r^{- 2} \end{matrix}

(Boxes 25.6 and 25.7). The scalar-wave potential, like the photon potential, is positive for all

r > 2 M

. It rises, from 0 at

r = 2 M

, to a barrier summit; then falls back to 0 at

r = \infty

.
(b) Show that there exist no physically acceptable, static scalar-wave perturbations of a Schwarzschild black hole. [More precisely, show that all static solutions to equation (32.27b) become infinite at either the horizon

(r = 2 M, r^{*} = - \infty)

or at radial infinity.] This suggests that somehow the black hole formed by collapse must divest itself of the star's external scalar field before it can settle down into a quiescent state.
(c) The general solution to the wave equation (32.27b) can be written in terms of a Fourier transform. For waves that begin near the horizon, propagate outward, and are partially transmitted and partially reflected ("rightward-propagating waves"), show that the general solution is

\begin{matrix} (32.28a) & Ψ_{ℓ} (t, r^{*}) = \int_{- \infty}^{\infty} A (k) R_{k}^{ℓ} (r^{*}) e^{- i k t} d k \end{matrix}

where

\begin{matrix} (32.28b) & d^{2} R_{k}^{t} / d r^{* 2} = [- k^{2} + V_{efr} (r)] R_{k}^{t}, \\ R_{k}^{t} = e^{i k r^{*}} + Γ_{k}^{(R)} e^{- i k r^{*}} as r^{*} ⟶ - \infty, \\ (32.28c) & R_{k}^{ℓ} = T_{k}^{(R)} e^{i k r^{*}} as r^{*} ⟶ \infty \end{matrix}

Show that the "reflection and transmission coefficients for rightward-propagating waves,"

Γ_{k}^{(R)}

and

T_{k}^{(R)}

, have the following asymptotic forms for

| k | ≪ 1 / M

(short wave number; long wavelength):

where

α

and

β

are constants of order unity. Give a similar analysis for waves that impinge on a Schwarzschild black hole from outside ("leftward-propagating waves").
(d) Show that, as the star collapses into the horizon, the world line of its surface in

(t, r^{*})

coordinates is

\begin{matrix} (32.29a) & r^{*} = R^{*} (t) \equiv - t - R_{0}^{*} \exp (- t / 2 M) + const. \end{matrix}

where

R_{0}^{*}

is related to the magnitude

a

of the surface's 4 -acceleration (

a > 0

for outward 4 -acceleration) by

\begin{matrix} (32.29b) & R_{0}^{*} = (8 M / e) {1 + 16 M a [M a + {(M^{2} a^{2} + \frac{1}{8})}^{1 / 2}]} . \end{matrix}

Thus, the world line of the surface appears to become null near the horizon

(t + r^{*} \equiv \tilde{V} =

constant); of course, this is due to pathology of the coordinate system there. Show, further, that the scalar field on the star's surface

(\vec{V} =

constant

)

must vary as

\begin{matrix} (32.29c) & Ψ_{t} = Q_{t 0} + Q_{t 1} e^{- \tilde{U} / 4 M}, \tilde{U} \equiv t - r^{*} \end{matrix}

when the star is approaching the horizon

(t ⟶ \infty, r^{*} ⟶ - \infty, \tilde{U} ⟶ \infty)

, in order that the rate of change of

Ψ_{ℓ}

be finite as measured on the star's surface. Notice that

Q_{10}

is the "final value" of the scalar field on the star's surface. It can be regarded as an outgoing wave with zero wave number (infinite wavelength); and, consequently, it gets completely and
destructively reflected by the effective potential [see equation (32.28d); also Box 32.2]. Conclusion: All multipoles of the scalar field die out at finite

r^{*}

t ⟶ \infty

. (Price's theorem for a scalar field.) For a more detailed analysis, including the rates at which the multipoles die out, see Price (1971, 1972a) or Thorne (1972).

Exercise 32.11. NEWMAN-PENROSE "CONSTANTS"

[See Press and Bardeen (1971), Bardeen and Press (1972), and Piir (1971) for more details than are presented here.]
Wheeler (1955) showed that Maxwell's equations for an

ℓ

-pole electromagnetic field residing in the Schwarzschild geometry can be reduced to the wave equation

\begin{matrix} (32.30) & - Ψ_{ℓ, t t} + Ψ_{i, r^{*} r^{*}} = (1 - \frac{2 M}{r}) \frac{ℓ (ℓ + 1)}{r^{2}} Ψ_{ℓ} \end{matrix}

[electromagnetic analogue of ( 32.27 b

)

]. After this equation has been solved, the components of the electromagnetic field can be obtained by applying certain differential operators to

Ψ_{l} (t, r^{*}) Y_{ℓ m} (θ, ϕ)

.
(a) Show that the general solution to the electromagnetic wave equation (32.30) for dipole

(ℓ = 1)

fields, with outgoing-wave boundary conditions at

r^{*} ⟶ + \infty

, has the form

\begin{matrix} (32.31a) & Ψ_{1} = f_{0} (\tilde{U}) + \frac{f_{1} (\tilde{U})}{r} + \frac{f_{2} (\tilde{U})}{r^{2}} + \dots, \end{matrix}

where

\begin{matrix} \tilde{U} = t - r^{*} is "retarded time", and \\ (32.31b) & f_{1}^{'} = f_{0}, f_{2}^{'} = 0, \dots, f_{n}^{'} = - \frac{(n + 1) (n - 2)}{2 n} f_{n - 1} + (n - 2) M f_{n - 2} \end{matrix}

When spacetime is flat

(M = 0)

, this solution becomes

\begin{matrix} (32.31~N) & Ψ_{1} = f_{1}^{'} (\tilde{U}) + f_{1} (\tilde{U}) / r \end{matrix}

[The

1 / r

fall-off of the radiation field

f_{1}^{'} (\tilde{U})

has been factored out of

Ψ_{1}

; see the scalar-wave function (32.27a).] The terms

f_{2} (\hat{U}) / r^{2} + \dots

, which are absent in flat spacetime, are attributable to backscatter of the outgoing waves by the curvature of spacetime. They are sometimes called the "tail" of the waves.
(b) Show that the general static dipole field has the form (32.31a) with

\begin{matrix} (32.32) & {(f_{0})}_{static} = 0; {(f_{1})}_{static} \equiv D = dipole moment; \\ {(f_{2})}_{static} = \frac{3}{2} M D; \dots \end{matrix}

t = 0

, let the star suddenly change its dipole moment to a new static value

D^{'}

. Equations (32.31b) demand that

f_{2}

be conserved ["Newman-Penrose (1965) constant"]. Hence,

f_{2}

will always exhibit a value,

_{2}^{3} M D

, corresponding to the old dipole moment; it can never change to

_{2}^{3} M D^{'}

. This is a manifestation of the tail of the waves that are generated by the sudden change in dipole moment. To understand this tail effect more clearly, and to discover an important flaw in the above result, evaluate the solution (32.31) for retarded time

\tilde{U} > 0

, using the assumptions
(1) field has static form (32.32) for

\tilde{U} < 0

,
(2)

f_{1} = D^{'}

for

\overset{⏜}{U} > 0

Put the answer in the form

Ψ_{1} = \frac{D^{'}}{r} + \frac{\frac{3}{2} M D}{r^{2}} + \sum_{n = 3}^{\infty} \frac{2 M (D^{'} - D) (- 1)^{n + 1} (n + 1) {\tilde{U}}^{n - 2}}{(2 r)^{n}} + O (\frac{M^{2}}{r^{3}}, \frac{M^{2} \tilde{U}}{r^{4}})

.
(The terms neglected are small compared to those kept for all

\tilde{U} / r

, so long as

r ≫

.) Evidently, so long as the series converges the Newman-Penrose "constant" (coefficient of

1 / r^{2}

) remembers the old

D

value and is conserved, as claimed above. Show, however, that the series diverges for

\tilde{U} > 2 r

-i.e., it diverges inside a sphere that moves outward with asymptotically

\frac{1}{3}

the speed of light. Thus, the Newman-Penrose "constant" is well-defined and conserved only outside the "

\frac{1}{3}

-speed-of-light cone."
(d) Sum the series in (32.34) to obtain a solution valid even for

\hat{U} > 2 r

\underset{\begin{matrix} new static \\ solution \end{matrix}}{\underset{⏟}{Ψ = \frac{D^{'}}{r} + \frac{3}{2} \frac{M D^{'}}{r^{2}}}} - \underset{"tail term"}{\underset{⏟}{\frac{2 M (D^{'} - D)}{r} \frac{(\tilde{U} + 3 r)}{(\tilde{U} + 2 r)^{2}}}} + O (\frac{M^{2}}{r^{3}})

= the series (32.34) for \tilde{U} < 2 r (domain of convergence of that series)

= \frac{D^{'}}{r} + \frac{3}{2} \frac{M D^{'}}{r^{2}} + O (\frac{M}{\tilde{U} r}, \frac{M^{2}}{r^{3}}) for \tilde{U} ≫ r ≫ M

From this result conclude that at fixed

r

and late times the electromagnetic field becomes asymptotically static, and the Newman-Penrose "constant" assumes the new value

_{2}^{3} M D

' appropriate to the new dipole moment.

сниетег 33

BLACK HOLES

A luminous star, of the same density as the Earth, and whose diameter should be two hundred and fifty times larger than that of the Sun, would not, in consequence of its attraction, allow any of its rays to arrive at us; it is therefore possible that the largest luminous bodies in the universe may, through this cause, be invisible.
P. S. LAPLACE (1798)

A dialog explaining why black holes deserve their name

§33.1. WHY "BLACK HOLE"?

Sagredus. What is all this talk about "black holes"? When an external observer watches a star collapse, he sees it implode with ever-increasing speed, until the relativistic stage is reached. Then it appears to slow down and become "frozen," just outside its horizon (gravitational radius). However long the observer waits, he never sees the star proceed further. How can one reasonably give the name "black hole" to such a frozen object, which never disappears from sight?

Salvatius. Let us take the name "black hole" apart. Consider first the blackness. Surely nothing can be blacker than a black hole. The very redshift that makes the collapsing star appear to freeze also makes it darken and become black. In the continuum approximation, where one ignores the discreteness of photons, the intensity of the radiation received by distant observers decreases exponentially as time passes,

L \propto \exp (- t / 3 \sqrt{3} M)

, with an exceedingly short

e

-folding time

τ = 3 \sqrt{3} M = (2.6 \times 10^{- 5} \sec) (M / M_{⊙})

Within a fraction of a second, the star is essentially black. Discreteness of photons makes it even blacker. The number of photons emitted before the star crosses its horizon is finite, so the exponential decay cannot continue
For a more detailed exposition of the foundations of "black-hole physics," see DeWitt and DeWitt (1973).
forever. Eventually-only

10^{- 3} (M / M_{⊙})

seconds after the star begins to

\dim

(see exercise 32.2)-the last photon that will ever get out reaches the distant observers. Thereafter nothing emerges. The star is not merely "essentially black"; it is "absolutely black."

Sagredus. Agreed. But it is the word "hole" that concerns me, not "black." How can one possibly regard the name "hole" as appropriate for an object that hovers forever just outside its horizon. True, absence of light makes the object invisible. But couldn't one always see it by shining a flashlight onto its surface? And couldn't one always fly down to its surface in a rocket ship and scoop up a few of the star's baryons? After all, as seen from outside the baryons at its surface will never, never, never manage to fall into the horizon!

Salvatius. Your argument sounds persuasive. To test its validity, examine the collapse of a spherically symmetric system, using the ingoing Eddington-Finkelstein diagram of Figure 33.1. Let a family of external observers shine their flashlights onto the star's surface, as you have suggested. Let the surface of the star be carefully silvered so it reflects back all light that reaches it. Initially (low down in the spacetime diagram of Figure 33.1) the ingoing light beams

Figure 33.1.
Spherical gravitational collapse of a star to form a black hole, as viewed in ingoing Eddington-Finkelstein coordinates. The "surface of last influence,"

R

, is an ingoing null surface that intersects the horizon in coincidence with the surface of the collapsing star. After an external observer, moving forward in time, has passed through the surface of last influence, he cannot interact with and influence the star before it plunges through the horizon. Thus, one can think of the surface of last influence as the "birthpoint" of the black hole. Before passing through this surface, the external observer can say his flashlight is probing the shape of a collapsing star; afterwards, he can regard his signals as probes of a black hole. For further discussion, see text.
reach the star's surface and are reflected back to the flashlights with no difficulty. But there is a critical point-an ingoing radial null surface

X

beyond which reflection is impossible. Photons emitted inward along

X

reach the star just as it is passing through its horizon. After reflection these photons fly "outward" along the horizon, remaining forever at

r = 2 M

. Other photons, emitted inward after the flashlight has passed through

R

, reach the surface of the star and are reflected only after the star is inside its horizon. Such photons can never return to the shining flashlights. Once inside the horizon, they can never escape. Thus, the total number of photons returned is finite and is subject to the same blackness decay law as is the intrinsic luminosity of the star. Moreover, if the observers do not turn on their flashlights until after they pass through the null surface

R

, they can never receive back any reflected photons! Evidently, flashlights are of no help in seeing the "frozen star."
Sagredus. I cannot escape the logic of your argument. Nevertheless, seeing is not the only means of interacting with the frozen star. I have already suggested swooping down in a rocket ship and stealing a few baryons from its surface. Similarly, one might let matter fall radially inward onto the frozen star. When the matter hits the star's surface, its great kinetic energy of infall will be converted into heat and into outpouring radiation.
Salvatius. Thus it might seem at first sight. But examine again Figure 33.1. No swooping rocket ship and no infalling matter can move inward faster than a light ray. Thus, if the decision to swoop is made after the ship passes through the surface

R

, the rocket ship has no possibility of reaching the star before it plunges through the horizon; the rocket and pilot cannot touch the star, sweep up baryons, and return to tell their tale. Similarly, infalling matter to the future of

ℜ

can never hit the star's surface before passing through the horizon. The surface

R

is, in effect, a "surface of last influence." Once anybody or anything has passed through

R

, he or it has no possibility whatever of influencing or interacting with the star in any way before it plunges through the horizon. Thus, from a "causal" or "interaction" standpoint, the collapsing star becomes a hole in space at the surface

ℜ

. This hole is not black at first. Radiation from the collapsing star still emerges after

R

because of finite light-propagation times, just as radiation still reaches us today from the hot big-bang explosion of the universe. But if an observer at radius

r ≫ 2 M

waits

for rays emitted at

R \sim 3 M

to get back to observer), then he will see the newly formed hole begin to turn black; and within a time

Δ t \sim (10^{- 3}

seconds

) (M / M_{⊙})

thereafter, it will be completely black.
Sagredus. You have convinced me. For all practical purposes the phrase "black hole" is an excellent description. The alternative phrases "frozen star" and "collapsed star," which I find in the pre-1969 physics literature, emphasize an optical-illusion aspect of the phenomenon. Attention must be directed away from the star that created the black hole, because beyond the surface of last influence one has no means to interact with that star. The star is irrelevant
to the subsequent physics and astrophysics. Only the horizon and its external spacetime geometry are relevant for the future. Let us agree to call that horizon the "surface of a black hole," and its external geometry the "gravitational field of the black hole."

Salvatius. Agreed.

§33.2. THE GRAVITATIONAL AND ELECTROMAGNETIC FIELDS OF A BLACK HOLE

The collapse of an electrically neutral star endowed with spherical symmetry produces a spherical black hole with external gravitational field described by the Schwarzschild line element

\begin{matrix} (33.1) & d s^{2} = - (1 - 2 M / r) d t^{2} + \frac{d r^{2}}{1 - 2 M / r} + r^{2} (d θ^{2} + \sin^{2} θ d ϕ^{2}) \end{matrix}

The surface of the black hole, i.e., the horizon, is located at

r = 2 M =

(gravitational radius). Only the region on and outside the black hole's surface,

r \geq 2 M

, is relevant to external observers. Events inside the horizon can never influence the exterior.

The gravitational collapse of a realistic star (nonspherical, collapse with small but nonzero net charge of one sign or the other) produces a black hole somewhat different from the simple Schwarzschild hole. For collapse with small charge and small asymmetries, perturbation-theory calculations (Box 32.2) predict a final black hole with external field determined entirely by the mass

M

, charge

Q

, and intrinsic angular momentum

S

of the collapsing star. For fully relativistic collapse, with large asymmetries and possibly a large charge, the final black hole (if one forms) is also characterized uniquely by

M, Q

, and

S

. This is the conclusion that strongly suggests itself in 1972 from a set of powerful theorems described in Box 33.1.
Why

M, Q

, and

S

should be the complete governors of the final external field of the black hole, one can understand heuristically as follows. Of all quantities intrinsic to any isolated source of gravity and electromagnetism, only

M, Q

, and

S

possess (and are defined in terms of) unique, conserved imprints in the distant external fields of the source (conserved Gaussian flux integrals; see Box 19.1 and §20.2). When a star collapses to form a black hole, its distant external fields are forced to maintain unchanged the imprints of

M, Q

, and

S

. In effect,

M, Q

, and

S

provide anchors or constraints on the forms of the fields. Initially other constraints are produced by the distributions of mass, momentum, stress, charge, and current inside the star. But ultimately the star plunges through a horizon, cutting itself off causally from the external universe. (The nonpropagation of long-wavelength waves through curved spacetime plays a key role in this cutoff; see Box 32.2.) Subsequently, the only anchors remaining for the external fields are the conserved imprints of

M

Q

, and

S

. Consequently, the external fields quickly settle down into unique shapes corresponding to the given

M, Q

, and

S

. Of course, the settling down involves dynamic changes of the fields and an associated outflow of gravitational and electro-

The structure of a black hole is determined uniquely by its mass

M

, charge

Q

, and intrinsic angular momentum,

S

Heuristic explanation of the

M - Q

-S uniqueness

Box 33.1 A BLACK HOLE HAS NO "HAIR"

The following theorems come close to proving that the external gravitational and electromagnetic fields of a stationary black hole (a black hole that has settled down into its "final" state) are determined uniquely by the hole's mass

M

, charge

Q

, and intrinsic angular momentum

S

-i.e., the black hole can have no "hair" (no other independent characteristics). For a detailed review, see Carter (1973).
I. Stephen Hawking (1971b, 1972a): A stationary black hole must have a horizon with spherical topology; and it must either be static (zero angular momentum), or axially symmetric, or both.
II. Werner Israel (1967a, 1968): Any static black hole with event horizon of spherical topology has external fields determined uniquely by its mass

M

and charge

Q

; moreover, those external fields are the Schwarzschild solution if

Q = 0

, and the Reissner-Nordstrøm solution (exercises 31.8 and 32.1) if

Q \neq 0

(both special cases of Kerr-Newman; see §33.2).
III. Brandon Carter (1970): "All uncharged, stationary, axially symmetric black holes with event horizons of spherical topology fall into disjoint families not deformable into each other. The black holes in each family have external gravitational fields determined uniquely by two parameters: the mass

M

and the angular momentum

S

." (Note: the "Kerr solutions"-i.e., "Kerr-Newman" with

Q = 0

-form one such family; it is very likely that there are no others, but this has not been proved as of December 1972. It is also likely that Carter's theorem can be extended to the case with charge; but this has also not yet been done.)
IV. Conclusions made by combining all three theorems:
(a) All stationary black holes are axially symmetric.
(b) All static (nonrotating) black holes are characterized uniquely by

M

and

Q

, and have the Reissner-Nordstrom form.
(c) All uncharged, rotating black holes fall into distinct and disjoint families, with each black hole in a given family characterized uniquely by

M

and

S

. The Kerr solutions form one such family. There may well be no other family.
V. Remarks and Caveats:
(a) The above statements of the theorems are all somewhat heuristic. Each theorem makes several highly technical assumptions, not stated here, about the global properties of spacetime. These assumptions seem physically reasonable and innocuous, but they might not be.
(b) Progress in black-hole physics is so rapid that, by the time this book is published, there may well exist theorems more powerful than the above, which really prove that "a black hole has no hair."
(c) For insight into the techniques of "global geometry" used in proving the above theorems and others like them, see Chapter 34 ; for greater detail see the forthcoming book by Hawking and Ellis (1973).
(d) For analyses which show that a black hole cannot exert any weak-interaction forces caused by the leptons which have gone down it, see Hartle

(1971, 1972)

and Teitelboim (1972b,c). For similar analyses which show absence of strong-interaction forces from baryons that have gone down the hole, see Bekenstein (1972a,b) and Teitelboim (1972a).
magnetic waves. And, of course, the outflowing waves carry off mass and angular momentum (but not charge), thereby leaving

M

and

S

changed. And, of course, the external fields must then readjust themselves to the new

M

and

S

. But the process will quickly converge, producing a black hole with specific final values of

M, Q

, and

S

and with external fields determined uniquely by those values.

The problem of calculating the external fields for given

M, Q

, and

S

and their given imprints, is analogous to the problem of Plateau-to calculate the shape of a soap film anchored to a wire of given shape.* One calculates the shape of the soap film by seeking a surface of minimum area spanning the bent wire. The condition of minimum area leads to a differential equation describing the soap film, which must be solved subject to the constraint imposed by the shape of the wire.

To calculate the external fields of a black hole, one can extremize the "action integral"

\int (R + E) \sqrt{- g} d^{4} x

for interacting gravitational and electromagnetic fields (see Chapter 21) subject to the anchored-down imprints of

M, Q

, and

S

at radial infinity, and subject to the existence of a physically nonsingular horizon (no infinite curvature at horizon!). Extremizing the action is equivalent to solving the coupled Einstein-Maxwell field equations subject to the constraints imprinted by

M, Q

, and

S

, and the existence of the horizon. The derivation of the solution and the proof of its uniqueness are much too complex to be given here. (See references cited in Box 33.1.) However, the solution turns out to be the "Kerr-Newman geometry" and its associated electromagnetic field.

†

Written in the

t, r, θ, ϕ

coordinates of Boyer and Lindquist (1967) (generalization of Schwarzschild coordinates), the Kerr-Newman geometry has the form

\begin{aligned} d s^{2} = & - \frac{Δ}{ρ^{2}} {[d t - a \sin^{2} θ d ϕ]}^{2} + \frac{\sin^{2} θ}{ρ^{2}} {[(r^{2} + a^{2}) d ϕ - a d t]}^{2} \\ (33.2) & + \frac{ρ^{2}}{Δ} d r^{2} + ρ^{2} d θ^{2}, \end{aligned}

where

\begin{matrix} (33.3a) & Δ \equiv r^{2} - 2 M r + a^{2} + Q^{2} \\ (33.3b) & ρ^{2} \equiv r^{2} + a^{2} \cos^{2} θ, \end{matrix}

\begin{matrix} (33.4) & a \equiv S / M \equiv angular momentum per unit mass. \end{matrix}

The corresponding electromagnetic field tensor, written as a 2 -form (recall:

d x^{α} \land

d x^{β} \equiv d x^{α} \otimes d x^{β} - d x^{β} \otimes d x^{α})

\begin{aligned} (33.5) & F = & Q ρ^{- 4} (r^{2} - a^{2} \cos^{2} θ) d r \land [d t - a \sin^{2} θ d ϕ] \\ + 2 Q ρ^{- 4} a r \cos θ \sin θ d θ \land [(r^{2} + a^{2}) d ϕ - a d t] . \end{aligned}

Variational principle for black-hole structure

Details of black-hole structure:
(1) metric ("Kerr-Newman geometry")
(2) electromagnetic field

Expressions (33.2) for the metric and (33.5) for the electromagnetic field are sufficiently long to be somewhat frightening. Therefore, it is helpful to develop some qualitative insight into them and into their implications before attempting detailed computations with them. Boxes 33.2, 33.3, and 33.4 develop qualitative insight by presenting, without derivation, a summary of the key features of the Kerr-Newman geometry and a summary of the physics and astrophysics of black holes. The remainder of this chapter is a Track-2 justification and derivation of some, but not all, of the results cited in Boxes 33.2-33.4.

Box 33.2 KERR-NEWMAN GEOMETRY AND ELECTROMAGNETIC FIELD

I. Equations for metric and electromagnetic field
A. Parameters appearing in equations:

M =

mass,

Q =

charge,

a \equiv S / M =

angular momentum per unit mass, all as measured by their standard imprints on the distant fields.
B. Constraint on parameters:

The Kerr-Newman geometry has a horizon, and therefore describes a black hole, if and only if

M^{2} \geq Q^{2} + a^{2}

. It seems likely that in any collapsing body which violates this constraint, centrifugal forces and/or electrostatic repulsion will halt the collapse before a size

\sim M

is reached; see equation (33.56).
C. Limiting cases:

Q = 0

Kerr (1963) geometry;

S = 0

Reissner-Nordstrom geometry and electromagnetic field

(exercises 31.8 and 32.1);

Q = S = 0

Schwarzschild geometry;

M^{2} = Q^{2} + a^{2}

"Extreme Kerr-Newman geometry."

D. Boyer-Lindquist (1967) coordinates

(t, r, θ, ϕ -

generalization of Schwarzschild coordinates; black hole rotates in

ϕ

direction):

\begin{aligned} (1) & d s^{2} = - (Δ / ρ^{2}) {[d t - a \sin^{2} θ d ϕ]}^{2} + (\sin^{2} θ / ρ^{2}) {[(r^{2} + a^{2}) d ϕ - a d t]}^{2} \\ + (ρ^{2} / Δ) d r^{2} + ρ^{2} d θ^{2}; \\ (2) & Δ \equiv r^{2} - 2 M r + a^{2} + Q^{2}, ρ^{2} \equiv r^{2} + a^{2} \cos^{2} θ . \\ F = \\ (3) & Q ρ^{- 4} (r^{2} - a^{2} \cos^{2} θ) d r \land [d t - a \sin^{2} θ d ϕ] \\ + 2 Q ρ^{- 4} a r \cos θ \sin θ d θ \land [(r^{2} + a^{2}) d ϕ - a d t] . \end{aligned}

E. Kerr coordinates

[\tilde{V}, r, θ, \tilde{ϕ}

-generalization of ingoing Eddington-Finkelstein coordinates;

(\tilde{V}, θ, \tilde{φ}) =

constant is an ingoing, "radial," null geodesic; black hole rotates in

\tilde{ϕ}

direction]:
Relationship to Boyer-Lindquist:

\begin{matrix} d \tilde{V} = d t + (r^{2} + a^{2}) (d r / Δ), \\ (4) & d \tilde{ϕ} = d ϕ + a (d r / Δ) \\ d s^{2} = - [1 - ρ^{- 2} (2 M r - Q^{2})] d {\tilde{V}}^{2} + 2 d r d \tilde{V} + ρ^{2} d θ^{2} \\ (5) & + ρ^{- 2} [{(r^{2} + a^{2})}^{2} - Δ a^{2} \sin^{2} θ] \sin^{2} θ d {\tilde{ϕ}}^{2} - 2 a \sin^{2} θ d \tilde{ϕ} d r \\ - 2 a ρ^{- 2} (2 M r - Q^{2}) \sin^{2} θ d \tilde{ϕ} d \tilde{V} . \\ (6) & F = Q ρ^{- 4} [(r^{2} - a^{2} \cos^{2} θ) d r \land d \tilde{V} - 2 a^{2} r \cos θ \sin θ d θ \land d \tilde{V} \\ - a \sin^{2} θ (r^{2} - a^{2} \cos^{2} θ) d r \land d \tilde{ϕ} + 2 a r (r^{2} + a^{2}) \cos θ \sin θ d θ \land d \tilde{ϕ}] . \end{matrix}

II. Properties of spacetime geometry
A. Symmetries (§33.4):

The metric coefficients in Boyer-Lindquist coordinates are independent of

t

and

ϕ

, and in Kerr coordinates are independent of

\tilde{V}

and

\tilde{ϕ}

. Thus the spacetime geometry is "time-independent" (stationary) and axially symmetric. The "Killing vectors" (

$ 25.2

) associated with these two symmetries are

(\partial / \partial t)_{r, θ, ϕ} = (\partial / \partial \tilde{V})_{r, θ, \tilde{ϕ}}

and

(\partial / \partial ϕ)_{t, r, θ} = (\partial / \partial \tilde{φ})_{\tilde{V}, r, θ}

.
B. Dragging of inertial frames and static limit (§33.4):

The "dragging of inertial frames" by the black hole's angular momentum produces a precession of gyroscopes relative to distant stars. By this precession one defines and measures the angular momentum of the black hole (see $§ $ 19.2$ and 19.3).
The dragging becomes more and more extreme the nearer one approaches the horizon of the black hole. Before the horizon is reached, at a surface described by

\begin{aligned} r & = r_{0} (θ) \\ (7) & \equiv M + \sqrt{M^{2} - Q^{2} - a^{2} \cos^{2} θ,} \end{aligned}

the dragging becomes so extreme that no observer can possibly remain at rest there (i.e., be "static") relative to the distant stars. At and inside this surface
(called the "static limit"), all observers with fixed

r

and

θ

must orbit the black hole in the same direction in which the hole rotates:

\begin{aligned} Ω \equiv d ϕ / d t \\ > \frac{a \sin θ - \sqrt{Δ}}{(r^{2} + a^{2}) \sin θ - \sqrt{Δ} a \sin^{2} θ} \\ (\geq 0 for a = S / M > 0 and r \leq r_{0}) . \end{aligned}

No matter how hard an observer, at fixed

(r, θ)

inside the static limit, blasts his rocket engines, he can never halt his angular motion relative to the distant stars.
3. The mathematical foundation for the above statement is this: world lines of the form

(r, θ, ϕ) =

constant [tangent vector

\propto \partial / \partial t =

"Killing vector in time direction"] change from being timelike outside the static limit to being spacelike inside it. Therefore, on and inside the static limit, no observer can remain at rest.
C. Horizon (§33.4):

The horizon is located at

\begin{matrix} (8) & r = r_{+} \equiv M + \sqrt{M^{2} - Q^{2} - a^{2}} \end{matrix}

As with the Schwarzschild horizon of a nonrotating black hole, so also here, particles and photons can fall inward through the horizon; but no particle or

Box 33.2 (continued)
photon can emerge outward through it.
3. The horizon is "generated" by outgoing null geodesics (outgoing photon world lines).
D. Ergosphere (§33.4):

The "ergosphere" is the region of spacetime between the horizon and the static limit. It plays a fundamental role in the physics of black holes (Box 33.3; §33.7).
The static limit and the horizon touch at the point where they are cut by the axis of rotation of the black hole $(θ = 0, π)$ ; they are well-separated elsewhere with the static limit outside the horizon, unless $a = 0$ (no rotation). When $a = 0$ , the static limit and horizon coincide; there is no dragging of inertial frames; there is no ergosphere.

Qualitative representation of horizon, ergosphere, and static limit [adapted from Ruffini and Wheeler (1971b)].
E. Singularity in Boyer-Lindquist coordinates:

For a nonrotating black hole, the Schwarzschild coordinates become singular at the horizon. One manifestation
of the singularity is the infinite amount of coordinate time required for any particle or photon to fall inward through the horizon, $t ⟶ \infty$ as $r ⟶ 2 M$ . One way to remove the singularity (Eddington-Finkelstein way) is to replace $t$ by a null coordinate

\tilde{V} = t + r + 2 M \ln | r / 2 M - 1 |

attached to infalling photons [so

(\partial / \partial r)_{\tilde{V}, θ, ϕ}

is vector tangent to photon world lines].
2. For a rotating black hole, the BoyerLindquist coordinates, being generalizations of the Schwarzschild coordinates, are also singular at the horizon. It requires an infinite coordinate time for any particle or photon to fall inward through the horizon,

t ⟶ \infty

r ⟶ r_{+}

. But that is not all. The dragging of inertial frames forces particles and photons near the horizon to orbit the black hole with

Ω \equiv d ϕ / d t > 0

. Consequently, for a particle falling through the horizon (

r ⟶ r_{+}

), just as

t ⟶ \infty

, so also

ϕ ⟶ \infty

(infinite twisting of world lines around horizon).
3. To remove the coordinate singularity, one must perform an infinite compression of coordinate time, and an infinite untwisting in the neighborhood of the horizon. Kerr coordinates achieve this by replacing

t

with a null coordinate

\tilde{V}

, and

ϕ

with an untwisted angular coordinate

\tilde{ϕ}

\begin{aligned} d \tilde{V} = d t + (r^{2} + a^{2}) (d r / Δ), \\ d \tilde{ϕ} = d ϕ + a (d r / Δ) \end{aligned}

Both of the new coordinates are attached to the world lines of a particular family of infalling photons;

(\partial / \partial r)_{\tilde{v}, θ, \tilde{ϕ}}

is the field of vectors tangent to the world lines of this family of photons (ingoing principal null congruence; §33.6).
F. Spacetime diagram:

A spacetime diagram in Kerr coordinates looks much like an EddingtonFinkelstein diagram for the Schwarzschild geometry. In both cases, one plots the surfaces of constant $\tilde{V}$ not as horizontal planes, but as "backward light cones" (" 45 -degree surfaces"), because they are generated by the world lines of ingoing photons. Equivalently, one plots surfaces of constant $\tilde{t} \equiv \hat{V} - r$ as horizontal planes.
The key differences between a Kerr diagram and an Eddington-Finkelstein diagram are: (a) Because the KerrNewman geometry is not spherical, a Kerr diagram with one rotational degree of freedom suppressed loses information about the geometry. Kerr diagrams are usually made for the equatorial "plane," $θ = π / 2$ . (b) Just as the horizon pulls the light cones inward, so the dragging of inertial frames tilts the light cones in the direction of increasing $\tilde{ϕ}$ , for $a > 0$ and $r =$ constant.
(c) The ingoing edge of a light cone $(d r / d \tilde{V} = - \infty)$ does not tilt toward increasing $\tilde{ϕ}$ ; the transformation from Boyer-Lindquist coordinates to Kerr coordinates untwists the tilt with decreasing $r$ , which would otherwise be produced by "frame dragging."
The shapes of the light cones reveal the special features of the static limit and horizon. At the static limit, a vertical world line $[r, θ, \tilde{ϕ}$ constant; $(\partial / \partial \tilde{V})_{r, θ, \tilde{Φ}}$ $= (\partial / \partial t)_{r, θ, ϕ} =$ tangent vector] lies on the light cone. At the horizon the light cones tilts fully inward, except for a single line of tangency to the horizon. Notice that the line of tangency has $d \hat{ϕ} / d \tilde{V} = a / (r_{+}^{2} + a^{2}) \neq 0$ . Equivalently, the outgoing null geodesics, which generate the horizon, twist about it ("barber-pole-twist")-yet another manifestation of the dragging of inertial frames.

Kerr diagram for equatorial slice (

θ = π / 2

) through the spacetime of an "extreme Kerr" black hole

(Q = 0, a = M)

View from above showing the shapes of the light cones as a function of radius

Box 33.2 (continued)

The Kerr diagram, like the EddingtonFinkelstein diagram, describes infall through the horizon in a faithful, nonsingular way.
[The term "Kerr diagram" is a misnomer. Kerr has not published such diagrams himself, though nowadays others construct such diagrams using his coordinate system. Penrose is the originator and greatest exploiter of such diagrams (see, e.g., Penrose, 1969). But several other types of diagrams bear Penrose's name, so it would be confusing to name them all after him.]
G. Maximal analytic extension of Kerr-Newman geometry:
When one abstracts the Schwarzschild geometry away from all sources (Chapter 31), one discovers that it describes an expanding and recontracting bridge, connecting two different universes. But in the context of black holes, only half of the Schwarzschild geometry (regions I and II) is relevant. The other half (regions III and IV) gets fully replaced by the interior of the star that collapsed to form the black hole. Because only a

part of the Schwarzschild geometry comes into play, ingoing EddingtonFinkelstein coordinates-which describe I and II well, but III and IV badly-are well-suited to black-hole physics.
Similarly, when one abstracts the KerrNewman geometry away from all sources, one discovers that it describes a much larger, and more complex spacetime manifold than one might ever have suspected. This "maximum analytic extension" of the Kerr-Newman geometry has been analyzed in detail by Boyer and Lindquist (1967) and by Carter (1966a, 1968a). But it is totally irrelevant to the subject of black holes, for two reasons. First, as with Schwarzschild, the star that collapsed to form the black hole replaces most of the inward extension of the Kerr-Newman manifold. Second, even outside the star, the Kerr-Newman geometry does not properly represent the true geometry at early times. At early times the star has not got far down the road to collapse. Gravitational moments of the star arise from mountains or prominences or turbulence or other particularities that have not yet gone into the meat grinder. The geometry departs from flatness (1) by a term that varies for large distances as mass divided by distance, and (2) by another term that varies as angular momentum divided by the square of the distance and multiplied by a spherical harmonic of order one, but also (3) by higher-order terms proportional to higher-order mass moments multiplied by higher spherical harmonics. These higher-order terms normally will deviate at early times from the corresponding terms in the mathematical analysis of the Kerr-Newman geometrythough the deviations will die out as time passes. For a system endowed with spherical symmetry, no such higherorder terms do occur or can occur. Therefore the geometry outside is

Schwarzschild in character at all stages of the collapse. However, when the system lacks spherical symmetry, the geometry outside initially departs from Kerr-Newman character. Only well after the collapse occurs (asymptotic future), and in the region at and outside the horizon, is the Kerr-Newman geometry a faithful descriptor of a black hole. This region is described in a nonsingular manner by Kerr coordinates and Kerr diagrams; and it is the only region that this book will explore.
H. Test-particle orbits

See

§ § 33.5 - 33.8

and Box 33.5 .
III. Properties of electromagnetic field (§33.3):
A. Far from the black hole, where spacetime is nearly flat, in the usual spherical orthonormal frame

(ω^{\hat{t}} = d t, ω^{\hat{ı}} = d r

ω^{\hat{θ}} = r d θ, ω^{\hat{ϕ}} = r \sin θ d ϕ)

, the electric and magnetic fields have dominant components

\begin{aligned} E_{\hat{r}} = \frac{Q}{r^{2}} \\ B_{\hat{r}} = \frac{2 Q a}{r^{3}} \cos θ, B_{\hat{θ}} = \frac{Q a}{r^{3}} \sin θ . \end{aligned}

These reveal that

Q =

charge of black hole,

R \equiv Q a =

magnetic dipole moment of black hole.
B. Notice that the gyromagnetic ratio,

γ \equiv

(magnetic moment)/(angular momentum), is equal to

Q / M =

(charge/mass), just as for an electron!
C. Notice that the value of the magnetic moment, like all other features of the black hole, is determined uniquely by the hole's mass, charge, and angular momentum:

π = Q S / M

. This illustrates the theorem (Box 33.1) that a black hole has no "hair."
D. Other electric and magnetic moments are nonzero, but are determined uniquely by

M, S

, and

Q

.
E. Near the black hole, the curvature of spacetime deforms the electric and magnetic fields produced by the charged, rotating black hole. For a mathematical description of this deformed field, see Cohen and Wald (1971); for a diagrammatic representation, Hanni and Ruffini (1973).

Box 33.3 THE ASTROPHYSICS OF BLACK HOLES

Black holes in nature should participate in astrophysical processes that are as varied as those for stars. By searching for observable phenomena associated with these processes, astronomers have a good chance of discovering the first black hole sometime during the 1970's. This box lists some possible astrophysical processes, and a few relevant references.

I. Mechanisms of Formation

A. "Direct, in isolation": A massive star

(M \geq

3 M_{⊙}

) collapses, almost spherically, producing a collapsed neutron-star core that is too massive to support itself against gravity. Gravity pulls the core on inward, producing a horizon and black hole. [May

Box 33.3 (continued)

and White (1966, 1967); Chapter 32 of this book.]
B. "Indirect, in isolation": "Collapse, pursuit, and plunge scenario" depicted in Figure 24.3 [Ruffini and Wheeler (1971b).]
C. "In the thick of things": Stars collected into a dense cluster (e.g., the nucleus of a galaxy) exchange energy. Some acquire energy and move out into a halo. Others lose energy and make a more compact cluster. This process of segregation continues. The cluster becomes so compact that collisions ensue and gas is driven off. The gas moves toward the center of the gravitational potential well. Out of it new stars form. The process continues. Eventually star-star collisions may become sufficiently energetic and inelastic that the centers of the colliding stars coalesce. In this way supermassive objects may be built up and may evolve. Ultimately (1) many "small" stars may collapse to form "small" black holes

(M \sim M_{⊙}); (2)

one or more supermassive stars may collapse to form huge black holes (

M \sim 10^{4} M_{⊙}

10^{9} M_{⊙}

); (3) the entire conglomerate of stars and gas and holes may become so dense that it collapses to form a single gigantic hole. [Sanders (1970), Spitzer (1971), LyndenBell (1967, 1969), Colgate (1967),

$ $ 24.5

, 24.6, 25.7 of this book.]
D. "Primordially": Perturbations in the initial density distribution of the expanding universe may produce collapse, resulting in "primordial black holes." Those holes would subsequently grow by accretion of radiation and matter. By today all such holes might have grown into enormous objects

[M \sim 10^{17} M_{⊙};

Zel'dovich and Novikov (1966)]; but some of them might have avoided such growth and might be as small as

10^{- 5}

grams [Hawking (1971a)].
II. How many black holes are there in our galaxy today?
Peebles (1972) has given an excellent review of this issue and of prospects for finding black holes in the near future. He says "a good fraction of the mass of the disc of our galaxy was deposited [long ago] in stars capable of collapsing to black holes. ... The indication is that the galaxy's disk may contain on the order of

10^{9}

black holes."
III. "Live" black holes versus "dead" black holes
A. A Schwarzschild black hole is "dead" in the sense that one can never extract from it any of its mass-energy. One aspect of this "deadness"-the fact that a Schwarzschild black hole is stable against small perturbations-is essential (1) to the identification of a black hole with the ultimate "ground state" of a large mass, and (2) to any assertion that general relativity theory predicts the possible existence of black holes. [For a proof of stability see Vishveshwara (1970). The problem was formulated, and most of the necessary techniques developed, by Regge and Wheeler (1957), with essential contributions also by Zerilli (1970a).] Thus a small pulse of gravitational (or other) radiation impinging on a Schwarzschild black hole does not initiate a transition of the black hole into a very different object or state.
B. A Kerr-Newman black hole-which is rotating or charged or both-is not dead. The rotational and electromagnetic contributions to the mass-energy can be extracted. (See

§ § 33.7

and 33.8 for mathematical details.) Thus, such black holes are "live"; they can inject energy into their surroundings. By a suitable arrangement of external apparatus, one can trigger an exponentially growing energy release [Press and Teukolsky (1972).] But for a perturbed
black hole in isolation, the release is always "controlled" and damped; i.e., Kerr black holes are stable in any classical context [Press and Teukolsky (1973)].
C. Most objects (massive stars; galactic nuclei; ...) that can collapse to form black holes have so much angular momentum that the holes they produce should be "very live" (

a

nearly equal to

M; S

nearly equal to

M^{2}

). [Bardeen (1970a).]
D. By contrast, it is quite probable (but far from certain) that no black hole in the universe has substantial charge-i.e., that all black holes have

Q ≪ M

. A black hole with

Q \sim M

(say,

Q > 0

for concreteness) would exert attractive electrostatic forces on electrons, and repulsive electrostatic forces on protons, that are larger than the hole"s gravitational pull by the factor

\frac{(electrostatic force)}{(gravitational force)} = \frac{e Q}{μ M} \sim \frac{e}{μ} \sim 10^{20} .

Here

e

is the electron charge and

μ

is the electron (or proton) mass. Such huge differential forces are likely to pull in enough charge from outside the hole to neutralize it.
E. But one has learned from the "unipolar induction process" for neutron stars [Goldreich and Julian (1968)] that charge neutralization can sometimes be circumvented. Whether any black-hole process can possibly prevent neutralization one does not know in 1972.
IV. Interaction of a black hole with its environment
A. Gravitational pull: A black hole exerts a gravitational pull on surrounding matter and stars. The pull is indistinguishable, at radii

r ≫ M

, from the pull of a star with the same mass.
B. Accretion and emission of

x

-rays and

γ

rays: Gas surrounding a black hole gets
pulled inward and is heated by adiabatic compression, by shock waves, by turbulence, by viscosity, etc. Before it reaches the horizon, the gas may become so hot that it emits a large flux of

x

-rays and perhaps even

γ

-rays. Thus, accreting matter can convert a black hole into a glowing "white" body [for a review of the literature, see Novikov and Thorne (1973)]. Accretion from a nonrotating gas cloud tends to decrease the angular momentum of a black hole [preferential accretion of particles with "negative" angular momentum; Doroshkevich (1966), Godfrey (1970a)]. But the gas surrounding a hole is likely to be rotating in the same direction as the hole itself, and to maintain

S \sim M^{2}

[more precisely,

S \approx 0.998 M^{2};

Thorne (1973b)].
C. A lump of matter (an "asteroid" or a "planet" or a star) falling into a black hole should emit a burst of gravitational waves as it falls. The total energy radiated is

E \sim 0.01 μ (μ / M)

, where

μ

is the mass of the object. [Zerilli (1970b); Davis, Ruffini, Press, and Price (1971); Figure 36.2 of this book.]
D. An object in a stable orbit around a black hole should spiral slowly inward because of loss of energy through gravitational radiation, until it reaches the most tightly bound, stable circular orbit. It should then fall quickly into the hole, emitting a "lastgasp burst" of waves. The total energy radiated during the slow inward spiral is equal to the binding energy of the last stable circular orbit:

E_{radiated} = μ - E_{last orbit}

= {\begin{cases} 0.0572 μ for Schwarzschild hole, \\ 0.4235 μ for Kerr hole with \end{cases}} S = M^{2}, Q = 0 . $

Here

μ

is the rest mass of the captured object. [Box 33.5.] The total energy in the last-gasp burst is

E \sim 0.01 μ (μ / M)

μ ≪ M

. [Fig. 36.2.]

Box 33.3 (continued)
E. When matter falls down a black hole, it can excite the hole's external spacetime geometry into vibration. The vibrations are gradually converted into gravitational waves, some of which escape, others go down the hole. [Press (1971, Goebel (1972).] These vibrations are analogous to an "incipient gravitational geon" [Wheeler (1962); Christodoulou (1971)]-except that for a vibrating black hole the background Kerr geometry holds the vibration energy together (prevents it from propagating away immediately), whereas in a geon it is curvature produced by the "vibration energy" itself that prevents disruption.
F. By a non-Newtonian, induction-zone (i.e., nonradiative) gravitational interaction, a black hole gradually transfers its angular momentum to any non-axially-symmetric, nearby distribution of matter or fields. [Hawking (1972a); Ipser (1971), Press (1972), Hawking and Hartle (1972).]
G. A star or planet falling into a large black hole will get torn apart by tidal gravitational forces. If the tearing occurs near but outside the horizon, it may eject a blob of stellar matter that goes out with relativistic velocity ("tube-of-toothpaste effect"). Moreover, the outgoing jet may extract a substantial amount of rotational energy from the hole's ergosphere-i.e., the hole might throw it off with a rest mass plus kinetic energy in excess of the rest mass of the original infalling object. [Wheeler (1971d); §§33.7 and 33.8.]
H. The magnetic field lines of a charged black hole may be anchored to surrounding plasma, may get wound up as the hole rotates, and may shake, twitch, and excite the plasma.
V. Collisions between black holes
A. Two black holes can collide and coalesce; but there is no way to blast a black hole apart into several black holes [Hawking (1972a); exercise 34.4].
B. When two black holes collide and coalesce, the surface area of the final black hole must exceed the sum of the surface areas of the two initial black holes ("second law of black-hole dynamics"; Hawking (1971a,b); Box 33.4; §34.5). This constraint places an upper limit on the amount of gravitational radiation emitted in the collision. For example, if all three holes are of the Schwarzschild variety and the two initial holes have equal masses

M / 2

, then

\begin{matrix} 4 π {(2 M_{final})}^{2} \geq 4 π [2 (M / 2)]^{2} + 4 π [2 (M / 2)]^{2}, \\ M_{final} \geq M / \sqrt{2}, \end{matrix}

so the energy radiated is

E_{radiated} \leq M - M / \sqrt{2} = 0.293 M

VI. Where and how to search for a black hole [For a detailed review, see Peebles (1971)]:
A. When it forms, by the burst or bursts of gravitational radiation given off during formation [Figure 24.3].
B. In a binary star system: black-hole component optically invisible, but may emit

x

-rays and

γ

-rays due to accretion; visible component shows telltale Doppler shifts [Hoyle, Fowler, Burbidge, and Burbidge (1964); Zel'dovich and Guseynov (1965); Trimble and Thorne (1969); Pringle and Rees (1972); Shakura and Sunyaev (1973)]. The velocity of the visible component and the period give information on the mass of the invisible component. If
mass of this invisible component is four solar masses or more, it cannot be an ordinary star, because an ordinary star of that mass would have

(4)^{3} = 64

times the luminosity of the sun. Neither can it be a white dwarf or a neutron star because either object, so heavy, would instantly collapse to a black hole. Therefore, it is attrac-tive-though not necessarily compelling [see Trimble and Thorne (1969)]-to identify the invisible object as a black hole.
C. [But one must not expect to see any noticeable gravitational lens action from a black hole in a binary system: if it taxed the abilities of astronomers for decades to see the black disc of Mercury,

4, 800 km

in diameter, swim across the great face of the sun, little hope there is to see a black hole with an effective radius of only

\sim 3

km , enormously more remote, occult a companion star. Significant lens action requires that the lens (black hole) be separated by a normal interstellar distance from the star it focuses; whence the impact parameter of the focused rays is more than a stellar radius, so the lens action is not more than that of a normal star. Moreover,
even with

10^{9}

black holes in the galaxy, only one per year would pass directly between the Earth and a more distant star, and produce significant lens action (Refsdal, 1964). Chance of watching the right spot on the sky at the right time with a sufficiently strong telescope: nil!]
D. At the center of a globular cluster, where a black hole may settle down, attract normal stars to its vicinity, and thereby produce a cusp in the distribution of light from the cluster. [Cameron and Truran (1971), Peebles (1971).]
E. In the nucleus of a galaxy, including even the Milky Way, where a single huge black hole (

M \sim 10^{4}

10^{8} M_{⊙}

) might sit as an end-product of earlier activity of the galactic nucleus. Such a hole will emit gravitational waves, light, and radio waves as it accretes matter. Much of the light may be converted into infrared radiation by surrounding dust. The black hole may also produce jets and other nuclear activity. [Lynden-Bell (1969), Lynden-Bell and Rees (1971), Wheeler (1971d), Peebles (1971).]

Box 33.4 THE LAWS OF BLACK-HOLE DYNAMICS

The black-hole processes described in Box 33.3 are governed by the standard laws of physics: general relativity, plus Maxwell electrodynamics, plus hydrodynamic, quantum mechanical, and other laws for the physics of matter and radiation. From these standard laws of physics, one can derive certain "rules" or "constraints," which all black-hole processes must satisfy. Those rules have a power, elegance, and simplicity that rival and resemble the power, elegance, and simplicity of the laws of thermodynamics. Therefore, they have been given the analogous name "the laws of black-hole dynamics" (Israel 1971). This box states two of the laws of black-hole

Box 33.4 (continued)

dynamics and some of their ramifications. Two additional laws, not discussed here, have been formulated by Bardeen, Carter, and Hawking (1973).
I. The First and Second Laws of Black-Hole Dynamics.
A. The first law.

Like the first law of thermodynamics, the first law of black-hole dynamics is the standard law of conservation of total energy, supplemented by the laws of conservation of total momentum, angular momentum, and charge. For detailed discussions of these conservation laws, see Box 19.1 and Chapter 20.
Specialized to the case where matter falls down a black hole and gravitational waves pour out, the first law takes the form depicted and discussed near the end of Box 19.1.
Specialized to the case of infalling electric charge, the first law says that the total charge $Q$ of a black hole, as measured by the electric flux emerging from it, changes by an amount equal to the total charge that falls down the hole,

Δ Q = q_{that falls in}

Specialized to the case where two black holes collide and coalesce (example given in Box 33.3), the first law says: (a) Let $P_{1}$ and $P_{2}$ be the 4-momenta of the two black holes as measured gravitationally, when they are so well-separated that they have negligible influence on each other. ( $P_{1}$ and $P_{2}$ are 4 -vectors in the surrounding asymptotically flat spacetime.) Similarly, let $J_{1}$ and $J_{2}$ be their total angular-momentum tensors (not intrinsic angular-momentum vectors!) relative to some arbitrarily chosen origin of coordinates, $P_{0}$ , in the surrounding asymptotically flat spacetime ( $J_{1}$ and $J_{2}$ contain orbital angular momentum, as well as intrinsic angular momentum; see Box 5.6.). (b) Let $P_{3}$ and $J_{3}$ be the similar total 4-momentum and angular momentum of the final black hole. (c) Let $P_{r}$ and $J_{r}$ be the total 4-momentum and angular momentum radiated as gravitational waves during the collision and coalescence. Then

P_{3} = P_{1} + P_{2} - P_{r}, J_{3} = J_{1} + J_{2} - J_{r}

[Note: to calculate the mass and intrinsic angular momentum of the final black hole from a knowledge of

P_{3}

and

J_{3}

, follow the prescription of Box 5.6. In that prescription, the world line of the final black hole is that world line, in the distant asymptotically Lorentz coordinates, on which the hole's distant spherical field is centered.
B. The second law [expounded and applied by Hawking (1971b, 1972a)]. When anything falls down a black hole, or when several black holes collide and coalesce or collide and scatter, or in any other process whatsoever involving black holes, the sum of the surface areas (or squares of "irreducible masses"-see equation 3 below) of all black holes involved can never decrease. (See

§ 34.5

for proof.) This is the second law of black-hole dynamics.
II. Reversible and Irreversible Transformations; Irreducible Mass [Christodoulou (1970); Christodoulou and Ruffini (1971)—results derived independently of and simultaneously with Hawking's discovery of the second law.]
A. Consider a single Kerr-Newman black hole interacting with surrounding matter and fields. Its surface area, at any moment of time, is given in terms of its momentary mass

M

, charge

Q

, and intrinsic angular momentum per unit mass

a \equiv S / M

\begin{matrix} (1) & A = 4 π [r_{+}^{2} + a^{2}] = 4 π [{(M + \sqrt{M^{2} - Q^{2} - a^{2}})}^{2} + a^{2}] \end{matrix}

(exercise 33.12). Interaction with matter and fields may change

M, Q

, and

a

in various ways;

M

can even be decreased-i.e., energy can be extracted from the black hole! [Penrose (1969); §33.7.] But whatever may be the changes, they can never reduce the surface area

A

. Moreover, if any change in

M, Q

, and

a

ever increases the surface area, no future process can ever reduce it back to its initial value.
B. Thus, one can classify black-hole processes into two groups.

Reversible transformations change $M, Q$ , or $a$ or any set thereof, while leaving the surface area fixed. They can be reversed, bringing the black hole back to its original state.
Irreversible transformations change $M, Q$ , or $a$ or any set thereof, and increase the surface area in the process. Such a transformation can never be reversed. The black hole can never be brought back to its original state after an irreversible transformation.
C. Examples of reversible transformations and of irreversible transformations induced by infalling particles are presented in $§ § 33.7$ and 33.8 .
D. The reversible extraction of charge and angular momentum from a black hole (decrease in $Q$ and $a$ holding $A$ fixed) necessarily reduces the black hole's mass (energy extraction!). By the time all charge and angular momentum have been removed, the mass has dropped to a final "irreducible value" of

\begin{matrix} (2) & M_{ir} = (A / 16 π)^{1 / 2} = (\binom{mass of Schwarzschild}{black hole of surface area A}) \end{matrix}

Box 33.4 (continued)
E. Expressed in terms of this final, irreducible mass, the initial mass-energy of the black hole (with charge

Q

and intrinsic angular momentum

S

) is

\begin{matrix} (3) & M^{2} = {(M_{ir} + \frac{Q^{2}}{4 M_{ir}})}^{2} + \frac{S^{2}}{4 M_{ir}^{2}} \end{matrix}

[This formula, derived by Christodoulou and Ruffini, may be obtained by combining equations (1), (2), and

S = M a]

.
F. Thus, one can regard the total mass-energy of a black hole as made up of an irreducible mass, an electromagnetic mass-energy, and a rotational energy. But one must resist the temptation to think of these contributions as adding linearly. On the contrary, they combine in a way [equation (3)] analogous to the way rest mass and linear momentum combine to give energy,

E^{2} =

m^{2} + p^{2}

.
G. Contours of constant

M / M_{i r}

are depicted below in the "charge-angular momentum plane." Black holes can exist only in the interior of the region depicted (

Q^{2} + a^{2} \leq M^{2}

). [Diagram adapted from Christodoulou (1971).]

H. Since a black hole's irreducible mass is proportional to the square root of its surface area, one can restate the second law of black-hole dynamics as follows:

In black-hole processes the sum of the squares of the irreducible masses of all black holes involved can never decrease.

§33.3. MASS, ANGULAR MOMENTUM, CHARGE, AND MAGNETIC MOMENT

It is instructive to verify that the constants

M, Q

, and

a

, which appear in equations (33.2)-(33.5) for the Kerr-Newman geometry and electromagnetic field, are actually the black hole's mass, charge, and angular momentum per unit mass, as claimed above.

Mass and angular momentum are defined by their imprints on the spacetime geometry far from the black hole. Therefore, to calculate the mass and angular momentum, one can expand the line element (33.2) in powers of

1 / r

and examine the leading terms:

\begin{aligned} d s^{2} = & - [1 - \frac{2 M}{r} + O (\frac{1}{r^{2}})] d t^{2} - [\frac{4 a M}{r} \sin^{2} θ + O (\frac{1}{r^{2}})] d t d ϕ \\ (33.6) & + [1 + O (\frac{1}{r})] [d r^{2} + r^{2} (d θ^{2} + \sin^{2} θ d {\dot{ϕ}}^{2})] \end{aligned}

The examination is facilitated by transforming to asymptotically Lorentz coordi-nates-

x = r \sin θ \cos ϕ, y = r \sin θ \sin ϕ, z = r \cos θ

\begin{aligned} d s^{2} = & - [1 - \frac{2 M}{r} + O (\frac{1}{r^{2}})] d t^{2} - [\frac{4 a M}{r^{3}} + O (\frac{1}{r^{4}})] [x d y - y d x] \\ (33.6') & + [1 + O (\frac{1}{r})] [d x^{2} + d y^{2} + d z^{2}] . \end{aligned}

Direct comparison with the "standard form" [equation (19.13)] of the metric far from a stationary rotating source reveals that (1) the parameter

M

is, indeed, the mass of the black hole; and (2) the intrinsic angular momentum vector of the black hole is

\begin{matrix} (33.7) & S = (a M) \partial / \partial z = (a M) \cdot (\binom{unit vector pointing along polar axis}{of Boyer-Lindquist coordinates}) . \end{matrix}

The charge is defined for the black hole, as for any source, by a Gaussian flux integral of its electric field over a closed surface surrounding the hole. The electric

The rest of this chapter is Track 2. To be prepared for it, one needs to have covered the Track-2 part of Chapter 32 (gravitational collapse). In reading it, one will be helped greatly by Chapter 25 (orbits in Schwarzschild geometry). The rest of this chapter is needed as preparation for Chapter 34 (singularities and global methods).

The metric far outside a black hole: imprints of mass and angular momentum

The electromagnetic field far outside a black hole:
(1) electric field
(2) magnetic field
(3) magnetic dipole moment

Nonspherical shape of hole's geometry
field in the asymptotic rest frame of the black hole has as its orthonormal components

\begin{matrix} E_{\hat{r}} = E_{r} = F_{r t} = Q / r^{2} + O (1 / r^{3}) \\ (33.8) & E_{\hat{θ}} = E_{θ} / r = F_{θ t} / r = O (1 / r^{4}) \\ E_{\hat{ϕ}} = E_{ϕ} / r \sin θ = F_{θ t} / r \sin θ = 0 \end{matrix}

Hence, the electric field is purely radial with a Gaussian flux integral of

4 π Q

, which reveals

Q

to be the black hole's charge.

A similar calculation of the dominant components of the magnetic field reveals

\begin{matrix} B_{\hat{r}} = F_{\hat{θ} \hat{ϕ}} = \frac{F_{θ ϕ}}{r^{2} \sin θ} = 2 \frac{Q a}{r^{3}} \cos θ + O (\frac{1}{r^{4}}) \\ (33.9) & B_{\hat{θ}} = F_{\hat{ϕ} \hat{r}} = \frac{F_{ϕ r}}{r \sin θ} = \frac{Q a}{r^{3}} \sin θ + O (\frac{1}{r^{4}}) \\ B_{\hat{ϕ}} = F_{\hat{r} \hat{θ}} = \frac{F_{r θ}}{r} = 0 \end{matrix}

This is a dipole magnetic field, and from it one immediately reads off the value

\begin{matrix} (33.10) & M = Q a = (Q / M) S = \underset{4 ["gyromagnetic ratio"]}{\underset{⏟}{(charge / mass)}} \times (angular momentum) \end{matrix}

for the magnetic moment of the black hole.
Just as the rotation of the black hole produces a magnetic field, so it also produces nonspherical deformations in the gravitational field of the black hole [see Hernandez (1967) for quantitative discussion]. But those deformations, like the magnetic moment, are not freely specifiable. They are determined uniquely by the mass, charge, and angular momentum of the black hole.

§33.4. SYMMETRIES AND FRAME DRAGGING

The metric components (33.2) of a Kerr-Newman black hole are independent of the Boyer-Lindquist time coordinate

t

and angular coordinate

ϕ

. This means (see §25.2) that

\begin{matrix} (33.11) & ξ_{(t)} \equiv (\partial / \partial t)_{r, θ, ϕ} and ξ_{(ϕ)} \equiv (\partial / \partial ϕ)_{t, r, θ} \end{matrix}

are Killing vectors associated with the stationarity (time-translation invariance) and axial symmetry of the black hole. The scalar products of these Killing vectors with themselves and each other are

\begin{matrix} (33.12a) & ξ_{(t)} \cdot ξ_{(t)} = g_{t t} = - (\frac{Δ - a^{2} \sin^{2} θ}{ρ^{2}}) = - (1 - \frac{2 M r - Q^{2}}{ρ^{2}}) \\ (33.12b) & ξ_{(t)} \cdot ξ_{(ϕ)} = g_{t ϕ} = \frac{a \sin^{2} θ (Δ - r^{2} - a^{2})}{ρ^{2}} = - \frac{(2 M r - Q^{2}) a \sin^{2} θ}{ρ^{2}} \\ (33.12c) & ξ_{(ϕ)} \cdot ξ_{(ϕ)} = g_{ϕ ϕ} = \frac{[{(r^{2} + a^{2})}^{2} - Δ a^{2} \sin^{2} θ] \sin^{2} θ}{ρ^{2}} \end{matrix}

Since Killing vectors are geometric properties of spacetime, with existence independent of any and all coordinate systems, their scalar products also have coor-dinate-free meaning. It so happens (not by chance, but by careful choice of coordinates!) that the Boyer-Lindquist metric components

g_{t t}, g_{t ϕ}

, and

g_{ϕ ϕ}

are equal to these coordinate-independent scalar products. Thus

g_{t t}, g_{t ϕ}

, and

g_{ϕ ϕ}

can be thought of as three scalar fields which embody information about the symmetries of spacetime. By contrast, the metric coefficients

g_{r r} = ρ^{2} / Δ

and

g_{θ θ} = ρ^{2}

carry no information at all about the symmetries.* They depend, for their existence and values, on the specific Boyer-Lindquist choice of coordinates.

Any observer who moves along a world line of constant

(r, θ)

with uniform angular velocity sees an unchanging spacetime geometry in his neighborhood. Hence, such an observer can be thought of as "stationary" relative to the local geometry. If and only if his angular velocity is zero, that is, if and only if he moves along a world line of constant

(r, θ, ϕ)

, will he also be "static" relative to the black hole's asymptotic Lorentz frame (i.e., relative to the "distant stars").

The precise definition of "angular velocity relative to the asymptotic rest frame" -or simply "angular velocity"-is

\begin{matrix} (33.13a) & Ω \equiv \frac{d ϕ}{d t} = \frac{d ϕ / d τ}{d t / d τ} = \frac{u^{ϕ}}{u^{t}} \end{matrix}

(see exercise 33.2). In terms of

Ω

, the Killing vectors, and the scalar products of Killing vectors, the 4-velocity of a stationary observer is

\begin{aligned} (33.13b) & u & = u^{t} (\partial / \partial t + Ω \partial / \partial ϕ) = \frac{ξ_{(t)} + Ω ξ_{(ϕ)}}{| ξ_{(t)} + Ω ξ_{(ϕ)} |} \\ = \frac{ξ_{(t)} + Ω ξ_{(ϕ)}}{{(- g_{t t} - 2 Ω g_{t ϕ} - Ω^{2} g_{ϕ ϕ})}^{1 / 2}} . \end{aligned}

A stationary observer is static if and only if

Ω

vanishes.
The stationary observers at given

r, θ

cannot have any and every angular velocity. Only those values of

Ω

are allowed for which the 4 -velocity

u

lies inside the future light cone-i.e., for which

{(ξ_{(t)} + Ω ξ_{(ϕ)})}^{2} = g_{t t} + 2 Ω g_{t ϕ} + Ω^{2} g_{ϕ ϕ} < 0

Stationary observers

Static observers

Angular velocity and 4-velocity of a stationary observer

Frame dragging, static limit, and ergosphere

Thus, the angular velocities of stationary observers are constrained by

\begin{matrix} (33.14) & Ω_{min} < Ω < Ω_{max}, \end{matrix}

where

\begin{aligned} (33.15a) & Ω_{min} & = ω - \sqrt{ω^{2} - g_{t t} / g_{ϕ ϕ}} \\ (33.15b) & Ω_{max} & = ω + \sqrt{ω^{2} - g_{t t} / g_{ϕ ϕ}} \\ (33.16) & ω \equiv \frac{1}{2} (Ω_{min} + Ω_{max}) & = - \frac{g_{ϕ t}}{g_{ϕ ϕ}} = \frac{(2 M r - Q^{2}) a}{{(r^{2} + a^{2})}^{2} - Δ a^{2} \sin^{2} θ} \end{aligned}

and it is assumed that

S / M = a > 0

. The following features of these limits are noteworthy. (1) Far from the black hole, one has

r Ω_{min} = - 1

and

r Ω_{max} = + 1

, corresponding to the standard limits imposed by the speed of light in flat spacetime.
(2) With decreasing radius,

Ω_{min}

increases ("dragging of inertial frames"). Finally, when

g_{t t}

reaches zero, i.e., at

\begin{matrix} (33.17) & r = r_{0} (θ) \equiv M + \sqrt{M^{2} - Q^{2} - a^{2} \cos^{2} θ} \end{matrix}

Ω_{min}

becomes zero. At and inside this surface, all stationary observers must orbit the black hole with positive angular velocity. Thus, static observers exist outside and only outside

r = r_{0} (θ)

. For this reason

r = r_{0} (θ)

is called the "static limit"; see Box 33.2. (3) As one moves through the static limit into the "ergosphere," one sees the allowed range of angular velocities become ever more positive (ever more "frame dragging"). At the same time, one sees the allowed range narrow down, until finally, at the horizon

\begin{matrix} (33.18) & r = r_{+} \equiv M + \sqrt{M^{2} - Q^{2} - a^{2}} \end{matrix}

the limits

Ω_{min}

and

Ω_{max}

coalesce

(ω^{2} = g_{t t} / g_{ϕ ϕ})

. Thus, at the horizon there are no stationary observers. All timelike world lines point inward. There is no escape from the black hole's "pull."

Further features of stationary observers and "frame dragging" are explored in the exercises.

EXERCISES

Exercise 33.1. KERR DESCRIPTION OF KILLING VECTORS

(a) Use the transformation law from Boyer-Lindquist coordinates to Kerr coordinates [equation (4) of Box 33.2] to show that

\begin{aligned} (33.19a) & ξ_{(t)} \equiv (\partial / \partial t)_{r, θ, ϕ} = (\partial / \partial \tilde{V})_{r, θ, \tilde{ϕ}} \\ (33.19b) & ξ_{(ϕ)} \equiv (\partial / \partial ϕ)_{t, r, θ} = (\partial / \partial \tilde{ϕ})_{\tilde{V}, r, θ} \end{aligned}

Verify explicitly by examining metric components that

\begin{matrix} (33.19c) & g_{\tilde{V} \tilde{V}} = g_{t t}, g_{\tilde{V} \tilde{ϕ}} = g_{t ϕ}, g_{\tilde{ϕ} \tilde{ϕ}} = g_{ϕ ϕ}, \end{matrix}

in accordance with equations (33.19a,b).
(b) Show that for a stationary observer (world line of constant

r, θ

), the angular velocity expressed in terms of Kerr coordinates is

Ω \equiv d ϕ / d t = d \tilde{ϕ} / d \tilde{\nabla} = u^{\tilde{ϕ}} / u^{\tilde{V}}

so that the entire discussion of stationary observers in terms of Kerr coordinates is identical to the discussion in terms of Boyer-Lindquist coordinates. Differences between the coordinate systems show up only when one moves along world lines of changing

r

. Reconcile this fact with the fact that both coordinate systems use the same coordinates

(r, θ)

but different time and aximuthal coordinates (

t, ϕ

versus

\tilde{V}, \tilde{ϕ}

Exercise 33.2. OBSERVATIONS OF ANGULAR VELOCITY

An observer, far from a black hole and at rest in the hole's asymptotic Lorentz frame, watches (with his eyes) as a particle moves along a stationary (nongeodesic) orbit near the black hole. Let

Ω = d ϕ / d t

be the particle's angular velocity, as defined and discussed above. The distant observer uses his stopwatch to measure the time required for the particle to make one complete circuit around the black hole (one complete circuit relative to the distant observer himself; i.e., relative to the hole's asymptotic Lorentz frame).
(a) Show that the circuit time measured is

2 π / Ω

. Thus,

Ω

can be regarded as the particle's "angular velocity as measured from infinity."
(b) Let the observer moving with the particle measure its circuit time relative to the asymptotic Lorentz frame, using his eyes and a stopwatch he carries. Show that his answer for the circuit time must be

\begin{matrix} (33.20) & Δ τ = \frac{2 π}{Ω} \underset{↑ ["redshift factor"]}{\underset{⏟}{{(- g_{t t} - 2 Ω g_{t ϕ} - Ω^{2} g_{ϕ ϕ})}^{1 / 2}}} \end{matrix}

Exercise 33.3. LOCALLY NONROTATING OBSERVERS

(Bardeen 1970b)

(a) Place a rigid, circular mirror ("ring mirror") at fixed

(r, θ)

around a black hole. Let an observer at

(r, θ)

with angular velocity

Ω

emit a flash of light. Some of the photons will get caught by the mirror and will skim along its surface, circumnavigating the black hole in the positive-

ϕ

direction. Others will get caught and will skim along in the negative-

ϕ

direction. Show that the observer will receive back the photons from both directions simultaneously only if his angular velocity is

Ω = ω (r, θ) = expression (33.16)

Thus in this case, and only in this case, can the observer regard the

+ ϕ

and

- ϕ

directions as equivalent in terms of local geometry. Put differently, in this case and only in this case is the observer "nonrotating relative to the local spacetime geometry." Thus, it is appropriate to use the name "locally nonrotating observer" for an observer who moves with the angular velocity

Ω = ω (r, θ)

.
(b) Associated with the axial symmetry of a black hole is a conserved quantity,

p_{ϕ} \equiv p \cdot ξ_{(ϕ)}

, for geodesic motion. This quantity for any particle-whether it is moving along a geodesic or not-is called the "component of angular momentum along the black hole's spin axis," or simply the particle's "angular momentum." (See

§ 33.5

below.) Show that of all stationary observers at fixed

(r, θ)

, only the "locally nonrotating observer" has zero angular momentum. [Note: Bardeen, Press, and Teukolsky (1972) have shown that the "locally nonrotating observer" can be a powerful tool in the analysis of physical processes near a black hole.]

Exercise 33.4. ORTHONORMAL FRAMES OF LOCALLY NONROTATING OBSERVERS

(a) Let spacetime be filled with world lines of locally nonrotating observers, and let each such observer carry an orthonormal frame with himself. Show that the spatial orientations of these frames can be so chosen that their basis 1 -forms are

\begin{matrix} (33.21) & \begin{array}{ll} ω^{\hat{t}} = {| g_{t t} - ω^{2} g_{ϕ ϕ} |}^{1 / 2} d t, & ω^{\hat{ϕ}} = {(g_{ϕ ϕ})}^{1 / 2} (d ϕ - ω d t), \\ ω^{\hat{r}} = (ρ / Δ^{1 / 2}) d r, & ω^{\hat{θ}} = ρ d θ . \end{array} \end{matrix}

More specifically, show that these 1 -forms are orthonormal and that the dual basis has

\begin{matrix} (33.22) & \partial / \partial \hat{t} = u \equiv 4 -velocity of locally nonrotating observer. \end{matrix}

Show that

u = - ω^{\hat{t}}

is a rotation-free field of 1 -forms

[d ω^{\hat{t}} \land ω^{\hat{t}} = 0

; exercise 4.4].
(b) One sometimes meets the mistaken notion that a "locally nonrotating observer" is in some sense locally inertial. To destroy this false impression, verify that: (i) such an observer has nonzero 4 -acceleration,

\begin{matrix} (33.23) & a = Γ_{\hat{j} i} e_{}} = \frac{1}{2} \nabla \ln | g_{t t} - ω^{2} g_{ϕ ϕ} | \end{matrix}

(ii) if such an observer carries gyroscopes with himself, applying the necessary accelerations at the gyroscope centers of mass, he sees the gyroscopes precess relative to his orthonormal frame (33.21) with angular velocity

\begin{aligned} Ω^{(precess)} & = Γ_{\hat{θ} \hat{ϕ} \hat{i}} e_{\hat{r}} + Γ_{\hat{δ} \hat{\hat{t}}} e_{\hat{θ}} \\ (33.24) & = \frac{1}{2} \frac{g_{ϕ ϕ}^{1 / 2}}{{| g_{t t} - ω^{2} g_{ϕ ϕ} |}^{1 / 2}} [\frac{ω_{, θ}}{ρ} e_{\hat{r}} - \frac{Δ^{1 / 2} ω_{, r}}{ρ} e_{\hat{θ}}] . \end{aligned}

[Hints: See exercise 19.2, equation (13.69), and associated discussions. The calculation of the connection coefficients is performed most easily using the methods of differential forms; see §14.6.]

Exercise 33.5. LOCAL LIGHT CONES

Calculate the shapes of the light cones depicted in the Kerr diagram for an uncharged (

Q = 0

) Kerr black hole (part II.F of Box 33.2). In particular, introduce a new time coordinate

\begin{matrix} (33.25) & \tilde{r} \equiv \tilde{V} - r \end{matrix}

for which the slices of constant

\tilde{t}

are horizontal surfaces in the Kerr diagram. Then the Kerr diagram plots

\tilde{t}

vertically,

r

radially, and

\tilde{ϕ}

azimuthally, while holding

θ = π / 2

("equatorial slice through black hole").
(a) Show that the light cone emanating from given

\tilde{t}, r, \tilde{ϕ}

has the form

\frac{d r}{d \tilde{t}} = a (\frac{d \tilde{ϕ}}{d \tilde{t}}) - \frac{2 M / r}{1 + 2 M / r} \pm \sqrt{\frac{1}{(1 + 2 M / r)^{2}} - \frac{r^{2} (d \tilde{ϕ} / d \tilde{t})^{2}}{1 + 2 M / r}}

(b) Show that the light cone slices through the surface of constant radius along the curves

\begin{matrix} (33.26b) & d r / d \tilde{t} = 0, d \tilde{ϕ} / d \tilde{t} = Ω_{min} and Ω_{max} \end{matrix}

where

Ω_{min}

and

Ω_{max}

are given by expressions (33.15a,b) (minimum and maximum allowed angular velocities for stationary observers).
(c) Show that at the static limit,

r = r_{0} (π / 2)

, the light cone is tangent to a curve of constant

r, θ, \tilde{ϕ}

.
(d) Show that the light cone slices the surface of constant

\tilde{ϕ}

along the curves

\begin{matrix} (33.26c) & \frac{d \tilde{ϕ}}{d \tilde{t}} = 0, \frac{d r}{d \tilde{t}} = - 1 and \frac{1 - 2 M / r}{1 + 2 M / r} . \end{matrix}

(e) Show that the light cone is tangent to the horizon.
(f) Make pictures of the shapes of the light cone as a function of radius.
(g) Describe qualitatively how the light cone must look near the horizon in Boyer-Lindquist coordinates. (Note: it will look "crazy" because the coordinates are singular at the horizon.)

§33.5. EQUATIONS OF MOTION FOR TEST PARTICLES [Carter (1968a)]

Let a test particle with electric charge

e

and rest mass

μ

move in the external fields of a black hole. Were there no charge down the black hole, the test particle would move along a geodesic (zero 4 -acceleration). But the charge produces an electromagnetic field, which in turn produces a Lorentz force on the particle:

μ a = e F \cdot u

. (Here

u

is the particle's 4 -velocity, and

a \equiv \nabla_{u} u

is its 4-acceleration.)

The geodesic equation,

a = 0

, for the uncharged case is equivalent to Hamilton's equations

\begin{matrix} (33.27a) & d x^{μ} / d λ = \partial K / \partial p_{μ}, d p_{μ} / d λ = - \partial K / \partial x^{μ} \end{matrix}

where

λ

is an affine parameter so normalized that

\begin{matrix} (33.27b) & d / d λ = p = 4 -momentum, \end{matrix}

and where

\begin{matrix} (33.27c) & K \equiv " super-Hamiltonian" = \frac{1}{2} g^{μ v} p_{μ} p_{v} \end{matrix}

(see exercise 25.2). Similarly (see exercise 33.6) the Lorentz-force equation,

μ a =

e F \cdot u

, for the charged case is equivalent to Hamilton's equations written in terms of position

x^{μ}

and "generalized momentum"

π_{μ}

\begin{matrix} (33.28a) & d x^{μ} / d λ = \partial H / \partial π_{μ}, d π_{μ} / d λ = - \partial H / \partial x^{μ} \end{matrix}

The form of the superhamiltonian

K

, in terms of the metric coefficients at the particle's location,

g^{μ ν} (x^{α})

, and the particle's charge

e

and generalized momentum

π_{μ}

, is

\begin{matrix} (33.28b) & H = \frac{1}{2} g^{μ ν} (π_{μ} - e A_{μ}) (π_{ν} - e A_{ν}) \end{matrix}

[See

§ 7.3

of Goldstein (1959) for the analogous superhamiltonian in flat spacetime.]

Superhamiltonian for a charged test particle in any electromagnetic field in curved spacetime

The first of Hamilton's equations for this superhamiltonian reduces to

\begin{matrix} (33.29a) & p^{μ} \equiv (4 -momentum) \equiv d x^{μ} / d λ = π^{μ} - e A^{μ} \end{matrix}

(value of

π^{μ}

in terms of

p^{μ}, e

, and

A^{μ}

); the second, when combined with the first, reduces to the Lorentz-force equation

\begin{matrix} (33.29b) & \begin{matrix} d p^{μ} / d λ + Γ^{μ}_{α β} p^{α} p^{β} = e F^{μ ν} p_{v} . \\ [p_{ν}, not u_{v} because \\ λ = τ / μ \end{matrix}] . \end{matrix}

For a Kerr-Newman black hole, the vector potential in Boyer-Lindquist coordinates can be put in the form

Vector potential for a charged black hole
"Constants of motion" for a charged test particle moving around a charged black hole:
(1) "energy at infinity"

E

(2) "axial component of angular momentum "

L_{z}

\begin{matrix} (33.30) & A = - \frac{Q r}{ρ^{2}} (d t - a \sin^{2} θ d ϕ) \end{matrix}

as one verifies by checking that

d A = \frac{1}{2} (A_{β, α} - A_{α, β}) d x^{α} \land d x^{β}

reduces to the Faraday 2 -form of equation (33.5).
There is good reason for going through all this formalism, rather than tackling head-on the Lorentz-force equation in its most elementary coordinate version,

\frac{d^{2} x^{α}}{d λ^{2}} + Γ^{α}_{β γ} \frac{d x^{β}}{d λ} \frac{d x^{γ}}{d λ} = e F^{α}_{β} \frac{d x^{β}}{d λ} .

The Hamiltonian formalism enables one to discover immediately two constants of the motion; the elementary Lorentz-force equation does not. The key point is that the components

A_{μ}

A

[equation (33.30)] and the components

g^{μ ν}

of the metric [inverse of

g_{μ ν}

of equation (33.2); see (33.35)] are independent of

t

and

ϕ

(stationarity and axial symmetry of both the electromagnetic field and the spacetime geometry). Consequently, the superhamiltonian is also independent of

t

and

ϕ

; and therefore Hamilton's equation

d π_{α} / d λ = - \partial K / \partial x^{α}

guarantees that

π_{t}

and

π_{ϕ}

are constants of the motion.
Far from the black hole, where the vector potential vanishes and the metric becomes

d s^{2} = - d t^{2} + d r^{2} + r^{2} (d θ^{2} + \sin^{2} θ d ϕ^{2})

the constants of the motion become

\begin{matrix} π_{t} = p_{t} = - p^{t} = - energy, \\ π_{ϕ} = p_{ϕ} = r p^{\hat{ϕ}} = (\binom{projection of angular momentum}{along black hole's rotation axis}) . \end{matrix}

Thus it is appropriate to adopt the names and notation

\begin{aligned} (33.31a) & E & \equiv ("energy at infinity") \equiv - π_{t} = - (p_{t} + e A_{t}) \\ (33.31b) & L_{z} & \equiv (\begin{matrix} "axial component of angular \\ momentum", or simply \\ "angular momentum" \end{matrix}) \equiv π_{ϕ} = p_{ϕ} + e A_{ϕ} \end{aligned}

for the constants of the motion

- π_{t}

and

π_{ϕ}

.
A third constant of the motion is the particle's rest mass

\begin{matrix} (33.31c) & μ = | p | = {(- g^{α β} p_{α} p_{β})}^{1 / 2} \end{matrix}

In general, four constants of the motion are needed to determine uniquely the orbit of a particle through four-dimensional spacetime. If the black hole were to possess an additional symmetry-e.g., if it were spherical, rather than merely axially symmetric-then automatically there would be a fourth constant of the motion. But in general, black holes are not spherical; so test-particle motion around a black hole possesses only three obvious constants. It is rather remarkable, then, that a constant turns out to exist. It was discovered by Carter (1968a), using Hamilton-Jacobi methods. As of 1973, nobody has given a cogent geometric explanation of why this fourth constant should exist-although hints of an explanation may be found in Carter (1968c) and Walker and Penrose (1970).

Carter's "fourth constant" of the motion, as derived in exercise 33.7, is

\begin{matrix} (33.31d) & Q = p_{θ}^{2} + \cos^{2} θ [a^{2} (μ^{2} - E^{2}) + \sin^{- 2} θ L_{z}^{2}] . \end{matrix}

The constant of the motion

\begin{matrix} (33.31e) & K \equiv Q + {(L_{z} - a E)}^{2} \end{matrix}

obtained by combining

Q, L_{z}

, and

E

, is often used in place of

Q

. Whereas

Q

can be negative,

K

is always nonnegative:

\begin{aligned} K & = p_{θ}^{2} + {(L_{z} - a E \sin^{2} θ)}^{2} / \sin^{2} θ + a^{2} μ^{2} \cos^{2} θ \\ \geq 0 everywhere \\ = 0 only for case of photon (μ = 0) moving along polar axis (θ = 0, π) \end{aligned}

The contravariant components of the test particle's 4-momentum,

p^{α} = d x^{α} / d λ

, are readily expressed in terms of the constants

E, L_{z}, μ, Q

, by combining equations (33.31) with the metric coefficients (33.2) and the components of the vector potential (33.30). The result is

\begin{matrix} (33.32a) & ρ^{2} d θ / d λ = \sqrt{Θ}, \\ (33.32b) & ρ^{2} d r / d λ = \sqrt{R}, \\ (33.32c) & ρ^{2} d ϕ / d λ = - (a E - L_{z} / \sin^{2} θ) + (a / Δ) P \\ (33.32d) & ρ^{2} d t / d λ = - a (a E \sin^{2} θ - L_{z}) + (r^{2} + a^{2}) Δ^{- 1} P \end{matrix}

Equations of motion for charged test particles

Here

ρ^{2} = r^{2} + a^{2} \cos^{2} θ

as defined in equation (33.3b), and the functions

Θ, R, P

are defined by

\begin{matrix} (33.33a) & Θ = Q - \cos^{2} θ [a^{2} (μ^{2} - E^{2}) + L_{z}^{2} / \sin^{2} θ], \\ (33.33b) & P = E (r^{2} + a^{2}) - L_{z} a - e Q r, \\ (33.33c) & R = P^{2} - Δ [μ^{2} r^{2} + {(L_{z} - a E)}^{2} + Q] . \end{matrix}

When working in Kerr coordinates (to avoid the coordinate singularity at the horizon), one must replace equations (33.32c) and (33.32d) by

\begin{matrix} (33.32c') & ρ^{2} d \tilde{V} / d λ = - a (a E \sin^{2} θ - L_{z}) + (r^{2} + a^{2}) Δ^{- 1} (\sqrt{R} + P), \\ (33.32d') & ρ^{2} d \tilde{ϕ} / d λ = - (a E - L_{z} / \sin^{2} θ) + a Δ^{- 1} (\sqrt{R} + P) \end{matrix}

[These follow from (33.32) and the transformation between the two coordinate systems-see equations (4) of Box 33.2.] In the above equations, the signs of

\sqrt{R}

and

\sqrt{Θ}

can be chosen independently; but once chosen, they must be used consistently everywhere.

Applications of these equations of motion will play a key role in the rest of this chapter.

EXERCISES

Exercise 33.6. SUPERHAMILTONIAN FOR CHARGED-PARTICLE MOTION

Show that Hamilton's equations (33.28a) for the Hamiltonian (33.28b) reduce to equation (33.29a) for the value of the generalized momentum, and to the Lorentz force equation (33.29b). [Hint: Use the relation

{(g^{α β} g_{β γ})}_{, μ} = 0.]

Exercise 33.7. HAMILTON-JACOBI DERIVATION OF EQUATIONS OF MOTION [Based on Carter (1968a)]

Derive the first-order equations of motion (33.32) for a charged particle moving in the external fields of a Kerr-Newman black hole. Use the Hamilton-Jacobi method [Boxes 25.3 and 25.4 of this book; also Chapter 9 of Goldstein (1959)], as follows.
(a) Throughout the superhamiltonian

K

of equation (33.28b), replace the generalized momentum

π_{α}

by the gradient

\partial S / \partial x^{α}

of the Hamilton-Jacobi function.
(b) Write down the Hamilton-Jacobi equation [generalization of equation (2) of Box 25.4] in the form

\begin{matrix} (33.34a) & - \frac{\partial S}{\partial λ} = H [x^{α}, \frac{\partial S}{\partial x^{β}}] = \frac{1}{2} g^{α β} (\frac{\partial S}{\partial x^{α}} - e A_{α}) (\frac{\partial S}{\partial x^{β}} - e A_{β}) . \end{matrix}

g^{α β}

for a Kerr-Newman black hole in Boyer-Lindquist coordinates are given by

\begin{aligned} g \equiv & g^{α β} \frac{\partial}{\partial x^{α}} \frac{\partial}{\partial x^{β}} = - \frac{1}{Δ ρ^{2}} {[(r^{2} + a^{2}) \frac{\partial}{\partial t} + a \frac{\partial}{\partial \dot{ϕ}}]}^{2} + \frac{1}{ρ^{2} \sin^{2} θ} {[\frac{\partial}{\partial ϕ} + a \sin^{2} θ \frac{\partial}{\partial t}]}^{2} \\ (33.35) & + \frac{Δ}{ρ^{2}} {(\frac{\partial}{\partial r})}^{2} + \frac{1}{ρ^{2}} {(\frac{\partial}{\partial θ})}^{2} . \end{aligned}

(d) Use these metric components and the components (33.30) of the vector potential to bring the Hamilton-Jacobi equation (33.33) into the concrete form

\begin{aligned} - \frac{\partial S}{\partial λ} = & - \frac{1}{2} \frac{1}{Δ ρ^{2}} {[(r^{2} + a^{2}) \frac{\partial S}{\partial t} + a \frac{\partial S}{\partial ϕ} - e Q r]}^{2} \\ (33.34b) & + \frac{1}{2} \frac{1}{ρ^{2} \sin^{2} θ} {[\frac{\partial S}{\partial ϕ} + a \sin^{2} θ \frac{\partial S}{\partial t}]}^{2} + \frac{1}{2} \frac{Δ}{ρ^{2}} {(\frac{\partial S}{\partial r})}^{2} + \frac{1}{2} \frac{1}{ρ^{2}} {(\frac{\partial S}{\partial θ})}^{2} \end{aligned}

(e) Solve this Hamilton-Jacobi equation by separation of variables. [Hint: Because the equation has no explicit dependence on

λ, ϕ

, or

t

, the solution must take the form

\begin{matrix} (33.36a) & S = \frac{1}{2} μ^{2} λ - E t + L_{z} ϕ + S_{r} (r) + S_{θ} (θ) \end{matrix}

where the values of the "integration constants" follow from

\partial S / \partial λ = - H, \partial S / \partial t = π_{t}

\partial S / \partial ϕ = π_{ϕ}

. Insert this assumed form into (33.35) and solve for

S_{r} (r)

and

S_{θ} (θ)

to obtain

\begin{matrix} (33.36b) & S_{r} = \int Δ^{- 1} \sqrt{R} d r, S_{θ} = \int \sqrt{Θ} d θ \end{matrix}

where

R (r)

and

Θ (θ)

are the functions defined in equation (33.33). Notice that the constant

Q

arises naturally as a "separation-of-variables constant" in this procedure. It was in this way that Carter originally discovered

Q

, following Misner's suggestion that he seek analogies to a constant in Newtonian dipole fields (Corben and Stehle, 1960, p. 209).]
(f) By successively setting

\partial S / \partial [Q + {(L_{z} - a E)}^{2}], \partial S / \partial μ^{2}, \partial S / \partial E

, and

\partial S / \partial L_{z}

to zero, obtain the following equations describing the test-particle orbits:

\begin{aligned} (33.37a) & \int^{θ} \frac{d θ}{\sqrt{Θ}} & = \int^{r} \frac{d r}{\sqrt{R}} \\ (33.37b) & λ & = \int^{θ} \frac{a^{2} \cos^{2} θ}{\sqrt{Θ}} d θ + \int^{r} \frac{r^{2}}{\sqrt{R}} d r \\ (33.37c) & t & = \int^{θ} \frac{- a (a E \sin^{2} θ - L_{z})}{\sqrt{Θ}} d θ + \int^{r} \frac{(r^{2} + a^{2}) P}{Δ \sqrt{R}} d r \\ (33.37~d) & ϕ & = \int \frac{- (a E \sin^{2} θ - L_{z})}{\sin^{2} θ \sqrt{Θ}} d θ + \int \frac{a P}{Δ \sqrt{R}} d r \end{aligned}

(g) By differentiating these equations and combining them, obtain the equations of motion (33.32) cited in the text.
(h) Derive equations (33.31) for

E, L_{z}, μ

, and

Q

by setting

\partial S / \partial x^{α} = π_{α} = p_{α} + e A_{α}

§33.6. PRINCIPAL NULL CONGRUENCES

Two special families of photon trajectories "mold themselves into" the Kerr-Newman geometry in an especially harmonious way. They are called the "principal null congruences" of the geometry. ("Congruence" is an elegant word that means "space-

Principal null congruences for the spacetime geometry of a black hole
filling family of curves.") These congruences are the solutions to the test-particle equations of motion (33.32) with

\begin{matrix} (33.38a) & μ = 0 (zero rest mass; photon), \\ (33.38b) & e = 0 (zero charge on photon), \\ (33.38c) & L_{z} = a E \sin^{2} θ (\begin{array}{l} a permissible value for L_{z} \\ only because d θ / d λ turns \\ out to be zero \end{array}), \\ (33.38d) & Q = - {(L_{z} - a E)}^{2} = - a^{2} E^{2} \cos^{4} θ . \end{matrix}

For these values of the constants of motion, the equations of motion (33.32) reduce to

\begin{matrix} (33.39a) & k^{θ} \equiv d θ / d λ = 0 \\ (33.39b) & k^{r} \equiv d r / d λ = \pm E \begin{matrix} (" + " for outgoing photons, \\ "-" for ingoing) \end{matrix} \\ k^{ϕ} \equiv d ϕ / d λ = a E / Δ \\ (33.39c) & k^{t} \equiv d t / d λ = (r^{2} + a^{2}) E / Δ \end{matrix}

Significance of the principal null congruences

In what sense are these photon trajectories more interesting than others? (1) They mold themselves to the spacetime curvature in such a way that, if

C_{α β γ δ}

is the Weyl conformal tensor (§13.5), and

^{*} C_{α β γ δ} = ϵ_{α β μ ν} C^{| μ ν |}_{γ δ}

is its dual, then

\begin{matrix} (33.40) & C_{α β γ [δ} k_{ϵ]} k^{β} k^{γ} = 0,^{*} C_{α β γ [δ} k_{c} k^{β} k^{γ} = 0 . \end{matrix}

[This relationship implies that the Kerr-Newman geometry is of "Petrov-Pirani type D" and that these photon trajectories are "doubly degenerate, principal null congruences." For details of the meanings and implications of these terms see, e.g., §8 of Sachs (1964), or Ehlers and Kundt (1962), or the original papers by Petrov (1954, 1969 ) and Pirani (1957).] (2) By suitable changes of coordinates (exercise 33.8), one can bring the Kerr-Newman metric into the form

\begin{matrix} (33.41) & d s^{2} = (η_{α β} + 2 H k_{α} k_{β}) d x^{α} d x^{β}, \end{matrix}

where

H

is a scalar field and

k_{α}

are the components of the wave vector for one of the principal null congruences (either one; but not both!). [This was the property of the Kerr-Newman metric that led to its original discovery (Kerr, 1963). For further detail on metrics of this form, see Kerr and Schild (1965).] (3) In Kerr coordinates (Box 33.2), the ingoing principal null congruence is

\begin{aligned} (33.42a) & r = - & E λ, θ = const, \tilde{ϕ} = const, \tilde{V} = const. \\ {\begin{matrix} arbitrary normalization \\ factor; can be removed \\ by redefinition of λ \end{matrix}] \end{aligned}

These ingoing photon world lines are the generators of the conical surface

\tilde{V} =

const. in the Kerr diagram of Box 33.2. (4) The only kind of particle that can remain forever at the horizon is a photon with world line in the outgoing principal null congruence (exercise 33.9). Such photon world lines are "generators" of the horizon (dotted curves with a "barber-pole twist" in Kerr diagram of Box 33.2). They have angular velocity

\begin{matrix} (33.42b) & Ω = \frac{d ϕ}{d t} = \frac{d \tilde{ϕ}}{d \tilde{V}} = \frac{a}{r_{+}^{2} + a^{2}} = \frac{a}{2 M^{2} - Q^{2} + 2 M {(M^{2} - a^{2} - Q^{2})}^{1 / 2}} \end{matrix}

Exercise 33.8. KERR-SCHILD COORDINATES

EXERCISES

(a) Show that in Kerr coordinates the ingoing null congruence (33.39) has the form (33.42a). Also show that the covariant components of the wave vector-after changing to a new affine parameter

λ_{new} = λ_{old} E

-are
(b) Introduce new coordinates

\tilde{t}, x, y, z

, defined by

\begin{matrix} (33.44a) & x + i y = (r + i a) e^{i \tilde{ϕ}} \sin θ, z = r \cos θ, \tilde{t} = \tilde{V} - r; \end{matrix}

and show that in this "Kerr-Schild coordinate system" the metric takes the form

\begin{matrix} (33.44b) & d s^{2} = (η_{α β} + 2 H k_{α}^{(in)} k_{β}^{(in)}) d x^{α} d x^{β}, \end{matrix}

where

\begin{matrix} (33.44c) & H = \frac{M r - \frac{1}{2} Q^{2}}{r^{2} + a^{2} (z / r)^{2}}, \\ (33.44d) & k_{α}^{(in)} d x^{α} = - \frac{r (x d x + y d y) - a (x d y - y d x)}{r^{2} + a^{2}} - \frac{z d z}{r} - d \bar{t} \end{matrix}

For the transformation to analogous coordinates in which

d s^{2} = (η_{α β} + 2 H k_{α}^{(out} k_{β}^{(out)}) d x^{α} d x^{β} .

see, e.g., Boyer and Lindquist (1967).

Exercise 33.9. NULL GENERATORS OF HORIZON

(a) Show that in Kerr coordinates the outgoing principle null congruence is described by the tangent vector

\begin{matrix} (33.45) & \frac{d θ}{d λ} = 0, \frac{d r}{d λ} = E, \frac{d \tilde{ϕ}}{d λ} = 2 a \frac{E}{Δ}, \frac{d \tilde{V}}{d λ} = 2 (r^{2} + a^{2}) \frac{E}{Δ} . \end{matrix}

(b) These components of the wave vector become singular at the horizon

(Δ = 0)

, not because of a singularity in the coordinate system-the coordinates are well-behaved!-but because of poor normalization of the affine parameter. For each outgoing geodesic, let

Δ_{0}

be a constant, defined as the value of

Δ

at the event where the geodesic slices the hypersurface

\tilde{V} = 0

. Then renormalize the affine parameter for each geodesic

\begin{matrix} (33.46) & λ_{new} = (E / Δ_{0}) λ_{old} \end{matrix}

Show that the resulting wave vectors

\begin{matrix} (') & \frac{d θ}{d λ} = 0, \frac{d r}{d λ} = Δ_{0}, \frac{d \tilde{ϕ}}{d λ} = 2 a \frac{Δ_{0}}{Δ}, \frac{d \tilde{ν}}{d λ} = 2 (r^{2} + a^{2}) \frac{Δ_{0}}{Δ} \end{matrix}

are well-behaved as one approaches the horizon; and show that the geodesics on the horizon have the form

\begin{matrix} (33.47) & θ = const., r = r_{+} = const., \tilde{ϕ} = 2 a λ, \tilde{V} = 2 (r_{+}^{2} + a^{2}) λ \end{matrix}

(c) Show that these are the only test-particle trajectories that remain forever on the horizon. [Hint: Examine the light cone.]

§33.7. STORAGE AND REMOVAL OF ENERGY FROM BLACK HOLES [Penrose (1969)]

When an object falls into a black hole, it changes the hole's mass, charge, and intrinsic angular momentum (first law of black-hole dynamics; Box 33.4). If the infalling object is large, its fall produces much gravitational and electromagnetic radiation. To calculate the radiation emitted, and the energy and angular momentum it carries away-which are prerequisites to any calculation of the final state of the black hole-is an enormously difficult task. But if the object is very small (size of object

<

size of horizon; mass of object

≪

mass of hole), and has sufficiently small charge, the radiation it emits in each circuit around the hole is negligible. For example, for gravitational radiation

\begin{matrix} (33.48) & \frac{(energy emitted per circuit)}{(rest mass of object)} \sim \frac{(rest mass of object)}{(mass of hole)} \end{matrix}

[see

§ 36.5

; also Bardeen, Press, and Teukolsky (1972)]. Because the energy emitted is negligible, radiation reaction is also negligible, and the object moves very nearly along a test-particle trajectory. In this case, application of the first law of black-hole dynamics is simple and straightforward.

Consider, initially, a small object that falls directly into the black hole from far away. According to the first law, it produces the following changes in the mass, charge, and angular momentum of the black hole:

\begin{matrix} (33.49a) & Δ M = E = ("energy at infinity" of infalling object) \\ (33.49b) & Δ Q = e = (charge of infalling object) \\ (33.49c) & Δ S \equiv Δ | S | = L_{z} = (\binom{component of object's angular momentum}{on black hole's rotation axis}) \end{matrix}

The infalling object will also change the direction of

S

. In the black hole's original asymptotic Lorentz frame, its initial angular momentum vector points in the

z

-direction,

{(S_{z})}_{initial} = S, {(S_{x})}_{initial} = 0, {(S_{y})}_{initial} = 0

Consequently, only the

z

-component of angular momentum of the infalling object can produce any significant change in the magnitude of

S

. But the

x

- and

y

-components,

L_{x}

and

L_{y}

, can change the direction of

S

. If the object has negligible speed at infinity, then it produces the changes (exercise 33.10):

\begin{matrix} (33.49~d) & Δ S_{x} = L_{x} = - (\sin ϕ_{\infty}) \sqrt{Θ_{\infty}} - (\cot θ_{\infty} \cos ϕ_{\infty}) L_{z} \\ (33.49e) & Δ S_{y} = L_{y} = (\cos ϕ_{\infty}) \sqrt{Θ_{\infty}} - (\cot θ_{\infty} \sin ϕ_{\infty}) L_{z} \\ (33.49f) & Δ {(S_{x}^{2} + S_{y}^{2})}^{1 / 2} = \sqrt{Q} = {(L^{2} - L_{z}^{2})}^{1 / 2} \end{matrix}

Here a subscript "

\infty

" means the value of a quantity at a point on the orbit far from the black hole (at "infinity").

Consider, next, a more complicated process, first conceived of by Penrose (1969): (1) Shoot a small object

A

into the black hole from outside with energy-at-infinity

E_{A}

, charge

e_{A}

, and axial component of angular momentum

L_{z A}

. (2) When the object is deep down near the horizon, let it explode into two parts,

B

and

C

, each of which subsequently moves along a new test-particle trajectory, with new constants of the motion

e_{B}

and

e_{C}, E_{B}

and

E_{C}, L_{z B}

and

L_{z C^{*}}

(3) So design the explosion that object

B

falls down the hole and gets captured, but object

C

escapes back to radial infinity. What will be the changes in mass, charge, and angular momentum of the black hole? According to the first law of black-hole dynamics,

\begin{aligned} Δ M & = (\begin{matrix} total energy that distant observers see \\ fall inward past themselves minus \\ total energy that they see reemerge \end{matrix}) \\ = E_{A} - E_{C} . \end{aligned}

Similarly,

Δ Q = e_{A} - e_{C}

and

Δ S = L_{z A} - L_{z C}

. Not unexpectedly, these changes can be written more simply in terms of the constants of motion for object

B

, which went down the hole. View the explosion "

A ⟶ B + C

" in a local Lorentz frame down near the hole, which is centered on the explosive event. As viewed in that frame, the explosion must satisfy the special relativistic laws of physics (equivalence principle!). In particular, it must obey charge conservation

\begin{matrix} (33.50a) & e_{A} = e_{B} + e_{C} \end{matrix}

and conservation of total 4-momentum

{(p_{A})}_{immediately before explosion} = {(p_{B} + p_{C})}_{immediately after explosion}

Moreover, conservation of 4-momentum

p

and charge

e

implies also conservation of generalized momentum

π \equiv p - e A

π_{A} = p_{A} - e_{A} A = p_{B} + p_{C} - (e_{B} + e_{C}) A = π_{B} + π_{C}

and hence also conservation of the components of generalized momentum along the vectors

\partial / \partial t

and

\partial / \partial ϕ

\begin{aligned} (33.50b) & E_{A} & \equiv - π_{t A} = - π_{t B} - π_{t C} = E_{B} + E_{C} \\ (33.50c) & L_{z A} & \equiv π_{ϕ B} + π_{ϕ C} = L_{z B} + L_{z C} \end{aligned}

(conservation of "energy-at-infinity" and "axial component of angular momentum" in explosion). Combining these conservation laws with the expressions

Δ M = E_{A} - E_{C}, Δ Q = e_{A} - e_{C}, and Δ S = L_{z A} - L_{z C}

one obtains

\begin{matrix} (33.51) & Δ M = E_{B}, Δ Q = e_{B}, Δ S = L_{z B} . \end{matrix}

This result restated in words: the changes in mass, charge, and angular momentum are equal to the "energy-at-infinity," charge, and "axial component of angular momentum" that object

B

carries inward across the horizon, even though

B

may have ended up on a test-particle orbit that does not extend back to radial infinity!
Straightforward extensions of the above thought experiment produce this generalization: In any complicated black-hole process that involves infalling, colliding, and exploding pieces of matter that emit negligible gravitational radiation, the total changes in mass, charge, and angular momentum of the black hole are

\begin{aligned} (33.52a) & Δ M & = (\begin{matrix} sum of values of energy-at-infinity, E, \\ for all objects which cross the horizon-with \\ E evaluated for each object at event of crossing \end{matrix}), \\ (33.52b) & Δ Q & = (\binom{similar sum, of charges, e, for}{all objects crossing horizon}), \\ (33.52c) & Δ S & = (\binom{similar sum of axial components of angular}{momentum, L_{z}, for all objects crossing horizon}) . \end{aligned}

This result is not at all surprising. It is precisely what one might expect from the most naive of viewpoints. Not so expected, however, is the following consequence [Penrose (1969)]: By injecting matter into a black hole in a carefully chosen way, one can decrease the total mass-energy of the black hole-i.e., one can extract energy from the hole.
For uncharged infalling objects, the key to energy extraction is the ergosphere [hence its name, coined by Ruffini and Wheeler (1971a) from the Greek word "

ϵ ρ γ o ν

" for "work"]. Outside the ergosphere, the Killing vector

ξ_{(t)} \equiv \partial / \partial t

is timelike, as is the 4-momentum

p

of every test particle; and therefore

E = - p \cdot ξ_{(t)}

is necessarily positive. But inside the ergosphere (between the horizon and the static limit),

ξ_{(t)}

Changes in

M, Q, S

for any process
nonradiative black-hole
is spacelike, so for certain choices of timelike momentum vector (certain orbits of uncharged test particles),

E = - p \cdot ξ_{(t)}

is negative, whereas for others it is positive. The orbits of negative

E

are confined entirely to the ergosphere. Thus, to inject an uncharged object with negative

E

into the black hole-and thereby to extract energy from the hole-one must always change its

E

from positive to negative and therefore also change its orbit, after it penetrates into the ergosphere. Of course, this is not difficult in principle-and perhaps not even in practice; see Figure 33.2.

For a charged object, electromagnetic forces alter the region where there exist orbits of negative energy-at-infinity. If the charges of object and hole have opposite sign, then the hole's electromagnetic field pulls inward on the object, giving it more kinetic energy when near the hole than one would otherwise expect. Thus,

- p \cdot ξ_{(t)}

becomes an overestimate of

E

\begin{matrix} (33.53) & E = - (p - e A) \cdot ξ_{(t)} = - p \cdot ξ_{(t)} + \underset{t [< 0 if e Q < 0]}{\underset{⏟}{e Q r / ρ^{2}}} \end{matrix}

and orbits with

E < 0

exist in a region somewhat larger than the ergosphere. If, on the other hand,

e

and

Q

have the same sign, then orbits with

E < 0

are confined to a region smaller than the ergosphere. For given values

e, Q

, and rest mass

μ

, the region where there exist orbits with

E < 0

is called the "effective ergosphere."

Exercise 33.10. ANGULAR MOMENTUM VECTOR FOR INFALLING PARTICLE

EXERCISE

Derive equations (

33.49 d, e, f

) for the components

L_{x}

and

L_{y}

of the orbital angular momentum of a particle falling into a black hole. Assume negligible initial speed,

E^{2} - μ^{2} \approx 0

§33.8. REVERSIBLE AND IRREVERSIBLE TRANSFORMATIONS [Christodoulou (1970), Christodoulou and Ruffini (1971)]

Take a black hole of given mass

M

, charge

Q

, and angular momentum

S

. By injection of small objects, make a variety of changes in

M, Q

, and

S

. Can one pick an arbitrary desired change,

Δ M, Δ Q

, and

Δ S

, and devise a process that achieves it? Or are there limitations?

The second law of black-hole dynamics (nondecreasing surface area of black hole; Box 33.4; proof in

$ 34.5

of next chapter) provides a strict limitation.

Then can all values within that limitation be achieved-and can that limitation be discovered by a direct examination of test-particle orbits?

The answer is yes; and, in fact, the limitation was discovered by Christodoulou (1970) and Christodoulou and Ruffini (1971) from an examination of test-particle orbits, independently of and simultaneously with Hawking's (1971) discovery of the second law of black-hole dynamics.

Figure 33.2 .
An advanced civilization has constructed a rigid framework around a black hole, and has built a huge city on that framework. Each day trucks carry one million tons of garbage out of the city to the garbage dump. At the dump the garbage is shoveled into shuttle vehicles which are then, one after another, dropped toward the center of the black hole. Dragging of inertial frames whips each shuttle vehicle into a circling, inward-spiraling orbit near the horizon. When it reaches a certain "ejection point," the vehicle ejects its load of garbage into an orbit of negative energy-at-infinity,

E_{garbage} < 0

. As the garbage flies down the hole, changing the hole's total mass-energy by

Δ M = E_{garbage ejected} < 0

, the shuttle vehicle recoils from the ejection and goes flying back out with more energy-at-infinity than it took down

\begin{aligned} E_{vehicle out} = & E_{vehicle + garbage down} - E_{garbage ejected} \\ > E_{vehicle + garbage down} \end{aligned}

The vehicle deposits its huge kinetic energy in a giant flywheel adjacent to the garbage dump; and the flywheel turns a generator, producing electricity for the city, while the shuttle vehicle goes back for another load of garbage. The total electrical energy generated with each round trip of the shuttle vehicle is

\begin{aligned} (Energy per trip) & = E_{vehicle out} - (rest mass of vehicle) \\ = (E_{vehicle + garbage down} - E_{garbage ejected}) - (rest mass of vehicle) \\ = (rest mass of vehicle + rest mass of garbage - Δ M) - (rest mass of vehicle) \\ = (rest mass of garbage) + (amount, - Δ M, by which hole's mass decreases) \end{aligned}

Thus, not only can the inhabitants of the city use the black hole to convert the entire rest mass of their garbage into kinetic energy of the vehicle, and thence into electrical power, but they can also convert some of the mass of the black hole into electrical power!

To derive the limitation of nondecreasing surface area from properties of testparticle orbits, one must examine what values of energy-at-infinity,

E

, are allowed at a given location

(r, θ)

outside a black hole. Equations (33.32a,b), when combined, yield the value of

E

in terms of a test particle's location

(r, θ)

, rest mass

μ

, charge

e

, axial component of angular momentum

L_{z}

, and momenta

p^{r} = d r / d λ, p^{θ} = d θ / d λ

in the

r

and

θ

directions:

\begin{matrix} (33.54a) & α E^{2} - 2 β E + γ = 0; E = \frac{β + \sqrt{β^{2} - α γ}}{α}, \end{matrix}

where

\begin{matrix} (33.54b) & α = {(r^{2} + a^{2})}^{2} - Δ a^{2} \sin^{2} θ > 0 everywhere outside horizon, \\ (33.54c) & β = (L_{z} a + e Q r) (r^{2} + a^{2}) - L_{z} a Δ, \\ (33.54d) & γ = {(L_{z} a + e Q r)}^{2} - Δ {(L_{z} / \sin θ)}^{2} - μ^{2} Δ ρ^{2} - ρ^{4} [{(p^{r})}^{2} + Δ {(p^{θ})}^{2}] . \end{matrix}

(One must take the positive square root,

+ \sqrt{β^{2} - α γ}

, rather than the negative square root; positive square root corresponds to 4 -momentum pointing toward future; while negative square root corresponds to past-pointing 4 -momentum; see Figure 33.3.)

Several features of the energy equation (33.54) are noteworthy. (1) For orbits in the equatorial "plane,"

θ = π / 2

and

p^{θ} \equiv 0

, the energy equation yields an effective potential for radial motion (Box 33.5). (2) Orbits of negative

E

must have

β < 0

and

γ > 0

-which can be achieved only if

L_{z} a < 0

and/or

e Q < 0

. Thus, one cannot decrease the mass of a black hole without simultaneously decreasing the magnitude of its charge or angular momentum or both. (3) For an orbit at given

(r, θ)

, with given

e

and

L_{z}, E

is a minimum if

p^{r} = p^{θ} = μ = 0

. Put differently, the rest mass and the

r

- and

θ

-components of momentum always contribute positively to

E

By injecting an object into a black hole, produce small changes

δ M = E, δ Q = e, δ S = L_{z},

in its mass, charge, and angular momentum. For given changes in

Q

and

S

, what range of changes in

M

is possible? Clearly

δ M

can be made as large as one wishes by making the rest mass

μ

sufficiently large. But there will be a lower limit on

δ M

. That limit corresponds to the minimum value of

E

for given

e

and

L_{z}

. The orbit of minimum

E

crosses the horizon (otherwise no changes in

M, Q, S

would occur!), so one can evaluate

E

there. At the horizon, as anywhere, a minimum for

E

is achieved if

μ = p^{r} = p^{θ} = 0

. Inserting these values and

r = r_{+}

(so

Δ = 0

) into equations (33.54), one finds

\begin{matrix} (33.55) & E_{min} = \frac{L_{z} a + e Q r_{+}}{r_{+}^{2} + a^{2}}, \end{matrix}

Properties of test-particle orbits:
(1)

E

as function of

μ, e, L_{z^{'}}

r, θ, p^{r}

(2) effective potential
(3) negative

E

requires

L_{z} a < 0

and/or

e Q < 0

Changes in black-hole properties due to injection of particles:

Figure 33.3.

Energy-at-infinity

E

allowed for a particle of angular momentum

L_{z}

and rest mass

μ

, which is (1) in the "equatorial plane"

θ = π / 2

, (2) at radius

r = 3 M / 2

, (3) of an uncharged (

Q = 0

) extreme-Kerr (

S = M^{2}

) black hole.

E

is here plotted against

L_{z}

. "Seas" of "positive and negative root" states are shown. The positive root states have energies-at-infinity given by equations (33.54)

E = \frac{β + \sqrt{β^{2} - α γ}}{α}

and have 4 -momentum vectors pointing into the future light cone. The negative root states (states of Dirac's "negative energy sea") have energies at infinity given by

E = \frac{β - \sqrt{β^{2} - α γ}}{α},

and have 4 -momentum vectors pointing into the past light cone. In the gap between the "seas" no orbits exist (forbidden region). The gap vanishes at the horizon

r = M

(infinite redshift of local energy gap,

2 μ

, gives zero gap in energy-at-infinity). [Figure adapted from Christodoulou (1971).]
corresponding to changes in the black-hole properties of

\begin{matrix} (33.56) & δ M \geq \frac{a δ S + r_{+} Q δ Q}{r_{+}^{2} + a^{2}} (\binom{absolute minimum value of}{δ M for given δ S and δ Q}) \end{matrix}

Notice an important consequence [Bardeen (1970a)]: if the black hole is initially of the "extreme Kerr-Newman" variety, with

M^{2} = a^{2} + Q^{2}

, so that one might fear a change which makes

M^{2} < a^{2} + Q^{2}

and thereby destroys the horizon, one's fears are unfounded. Equation (33.56) then demands (since

r_{+} = M

and

S = M a

)

M δ M \geq a δ a + Q δ Q

M^{2}

remains greater than or equal to

a^{2} + Q^{2}

, and the horizon is preserved.

Box 33.5 ORBITS OF TEST PARTICLE IN "EQUATORIAL PLANE" OF KERR-NEWMAN BLACK HOLE

Radial motion is governed by energy equation (33.54) with

θ = p^{θ} = 0

\begin{matrix} (1) & α E^{2} - 2 β E + γ_{0} - r^{4} {(p^{r})}^{2} = 0; E = \frac{β + \sqrt{β^{2} - α γ_{0} + α r^{4} {(p^{r})}^{2}}}{α} \end{matrix}

α, β, γ_{0}

are functions of

r

and of constants of motion,

\begin{matrix} (2a) & α = {(r^{2} + a^{2})}^{2} - Δ a^{2} > 0 \\ (2b) & β = (L_{z} a + e Q r) (r^{2} + a^{2}) - L_{z} a Δ \\ (2c) & γ_{0} = {(L_{z} a + e Q r)}^{2} - Δ L_{z}^{2} - μ^{2} r^{2} Δ \end{matrix}

p^{r} = (

radial momentum

)

\begin{matrix} (3) & p^{r} = d r / d λ \end{matrix}

Thus, equation (1) is an ordinary differential equation for

d r / d λ

.
Qualitative features of the radial motion can be read off an effective-potential diagram. The effective potential

V (r)

is the minimum allowed value of

E

at radius

r

V (r) = \frac{β + \sqrt{β^{2} - α γ_{0}}}{α}

As in the Schwarzschild case (Figure 25.2), the allowed regions for a particle of energy-at-infinity

E

are the regions with

V (r) \leq E

; and the turning points (

p^{r} =

d r / d λ = 0

) occur where

V (r) = E

.
Stable circular orbits occur at the minima of

V (r)

. By examining

V (r)

closely, one finds that for uncharged black holes the innermost stable circular orbit (most tightly bound orbit) has the characteristics here tabulated [table adapted from Ruffini and Wheeler (1971b)].

Characteristic

Newtonian (Figure 25.2)

Schwarzschild

\begin{aligned} (a = Q = 0) \\ (Figure 25.2) \end{aligned}

Extreme Kerr

(a^{2} = M^{2}, Q = 0)

(see figure)

[Bardeen (1970a)]

r / M

E / μ

- \infty

2 \sqrt{2} / 3

\sqrt{3}

5 / (3 \sqrt{3})

\begin{matrix} (μ - E) / μ = \\ "fractional \\ binding" \end{matrix}

+ \infty

0.0572

0.4226

0.0377

| L_{z} | / μ M

2 \sqrt{3}

\sqrt{3}

22/(3

\sqrt{3}

)

Box 33.5 (continued)

For a charged extreme Kerr-Newman black hole (

M^{2} = Q^{2} + a^{2}, Q \neq 0

and

a \neq 0

) stable circular orbits with 100 per cent binding

(E = 0)

are achieved in the limit

\frac{e}{μ} ⟶ - \infty, \frac{Q}{M} ⟶ 0 (so a ⟶ M), and (\frac{e}{μ}) \cdot (\frac{Q}{M}) ⟶ - \infty .

[Christodoulou and Ruffini (1971)].
The effective potential for an uncharged, extreme Kerr black hole

(a = M)

is shown in the figure [figure adapted from Ruffini and Wheeler (1971b)]. For detailed diagrams of orbits in the equatorial plane, see de Felice (1968). For many interesting properties of orbits that are not confined to the equatorial plane, see Wilkins (1972).

The general limit (33.56) on the change in mass can be rewritten in an alternative form [Christodoulou (1970), Christodoulou and Ruffini (1971)]:

\begin{matrix} (33.57) & δ M_{ir} \geq 0 \end{matrix}

where

\begin{matrix} (33.58) & M_{ir} \equiv \frac{1}{2} \sqrt{r_{+}^{2} + a^{2}} = \frac{1}{2} {[{(M + \sqrt{M^{2} - Q^{2} - a^{2}})}^{2} + a^{2}]}^{1 / 2} \end{matrix}

is the "irreducible mass" of the black hole. Equation (33.57) states that no black-hole transformation produced by the injection of small lumps of matter can ever reduce the irreducible mass of a black hole. This result is actually a special case of the second law of black-hole dynamics, since the surface area of a black hole is

\begin{matrix} (33.59) & A = 16 π M_{ir}^{2} \end{matrix}

(Exercise 33.12).
Equation (33.58) can be combined with

a = S / M

and inverted to yield

\begin{aligned} (33.60) & M^{2} = {(M_{ir} + \frac{Q^{2}}{4 M_{ir}})}^{2} + \frac{S^{2}}{4 M_{ir}^{2}} . \\ [\begin{matrix} irreducible con- \\ tribution to mass \end{matrix}] \begin{matrix} \frac{↑}{electromagnetic con-} \begin{matrix} tribution to mass \end{matrix} \\ [\begin{matrix} rotational con- \\ tribution to mass \end{matrix}] \end{matrix} \end{aligned}

A black-hole transformation that holds fixed the irreducible mass is reversible; one that increases it is irreversible. The derivation of equation (33.56) revealed that the only injection processes that actually achieve the minimum possible value for

δ M

(and thus make

δ M_{ir} = 0

) are those with

μ = p^{r} = p^{θ} = 0

at the horizon,

r = r_{+}

. Restated in words: To produce a reversible transformation by injecting an object into a black hole, one must (1) give the object a rest mass

μ

extremely small compared to its charge

ϵ

or axial component of angular momentum

L_{z}

μ / e ≪ 1 and / or μ^{2} / L_{z}^{2} ≪ 1

and (2) set the object down "extremely gently" (

p^{r} = p^{θ} = 0

), extremely close to the horizon

(r = r_{+})

. This does not sound too difficult until one recalls that objects with

p^{r} = p^{θ} = 0

at the horizon must be moving outward with the speed of light, and that the nearer one approaches the horizon as one sets down the object, the greater one's danger of "slipping" and getting swallowed!

Clearly, any actual injection process will depart somewhat from irreversibility. Reversibility is an idealized limit, approachable but not attainable.

EXERCISES Exercise 33.11. IRREDUCIBLE MASS IS IRREDUCIBLE

Show that condition (33.56) is equivalent to

δ M_{ir} \geq 0

Exercise 33.12. SURFACE AREA OF A BLACK HOLE

Show that the surface area of the horizon of the Kerr-Newman geometry [area of surface

r = r_{+}

and

t =

const (Boyer-Lindquist coordinates) or

\tilde{V} =

const (Kerr coordinates)

]

16 π M_{ir}^{2}

Exercise 33.13. ANGULAR VELOCITY OF A BLACK HOLE

A general theorem [Hartle (1970) for relativistic case; Ostriker and Gunn (1969) for nonrelativistic case] says that, if one injects angular momentum into a rotating star while holding fixed all other contributions to its total mass-energy (contributions from entropy and from baryonic rest mass), then the injection produces a change in total mass-energy given by

\begin{matrix} (33.61) & δ (mass-energy) = (\binom{angular velocity of star}{at point of injection}) δ (angular momentum) \end{matrix}

By analogy, if one injects an angular momentum

δ S

into a rotating black hole while holding fixed all other contributions to its total mass-energy (contributions from irreducible mass and from charge), one identifies the coefficient

Ω_{h}

in the equation

δ M = Ω_{h} δ S

as the angular velocity of the hole:

\begin{matrix} (33.62) & Ω_{h} = (\partial M / \partial S)_{Q, M_{i r}} \end{matrix}

(a) Show that the angular velocity of a black hole is equal to

\begin{matrix} (33.63) & Ω_{h} = \frac{a}{r_{+}^{2} + a^{2}} \end{matrix}

Notice that this is precisely the angular velocity of photons that live forever on the horizon [equation (33.42b); "barber-pole twist" of null generators of horizon].
(b) Show that any object falling into a black hole acquires an angular velocity (relative to Boyer-Lindquist coordinates) of

Ω = d ϕ / d t = Ω_{h}

in the late stages, as it approaches the horizon. (Recall that the horizon is a singularity of the Boyer-Lindquist coordinates. This is the reason that every object, regardless of its

L_{z}, E, e, μ, Q

, can approach and does approach

Ω = Ω_{h}

Exercise 33.14. SEPARATION OF VARIABLES FOR WAVE EQUATIONS

This chapter has studied extensively the motion of small objects in the external fields of black holes. Of almost equal importance, but not so well-understood yet because of its complexity, is the evolution of weak electromagnetic and gravitational perturbations ("waves") in the Kerr-Newman geometry. Just as one had no à priori reason to expect a "fourth constant" for test-particle motion in the Kerr-Newman geometry, so one had no reason to expect separability for Maxwell's equations, or for the wave equations describing gravitational perturbations-or even for the scalar wave equation

◻ ψ \equiv - ψ_{α}^{α} = 0

. Thus it came as a great surprise when Carter (1968c) proved separability for the scalar wave equation, and later when Teukolsky

(1972, 1973)

separated both Maxwell's equations and the wave equations for gravitational perturbations.

Show that separation of variables for the scalar-wave equation in the (uncharged) Kerr geometry yields solutions of the form

\begin{matrix} (33.64a) & ψ = {(r^{2} + a^{2})}^{- 1 / 2} u_{i m} (r) S_{m ℓ} (- i ω a, \cos θ) e^{i (m ϕ - ω t)} \end{matrix}

where

m

and

ℓ

are integers with

0 \leq | m | \leq ℓ; S_{m ℓ}

is a spheroidal harmonic [see Meixner and Schärfke (1954)]; and

u_{ℓ m}

satisfies the differential equation

\begin{matrix} (33.64b) & - d^{2} u / d r^{* 2} + V u = 0 \end{matrix}

In order to put the equation in this form, define a Regge-Wheeler (1957) "tortoise"-type radial coordinate

r^{*}

\begin{matrix} (33.64c) & d r^{*} = Δ^{- 1} (r^{2} + a^{2}) d r \end{matrix}

and find an effective potential

V (r^{*})

given by

\begin{aligned} V = & - {(ω - \frac{m a}{r^{2} + a^{2}})}^{2} + [(m - ω a)^{2} + Q] {(r^{2} + a^{2})}^{- 2} Δ \\ (33.64d) & + 2 (M r - a^{2}) {(r^{2} + a^{2})}^{- 3} Δ + 3 a^{2} {(r^{2} + a^{2})}^{- 4} Δ^{2} \end{aligned}

In this radial equation

Q

is a constant (analog of Carter's constant for particle motion), given in terms of

m

and

ℓ

\begin{matrix} (33.64e) & Q \equiv λ_{m ℓ} - m^{2}; λ_{m ℓ} = [\begin{array}{l} eigenfunction of spheroidal harmonic; \\ see Meixner and Schärfke (1954) \end{array}] . \end{matrix}

[These details of the separated solution were derived by Brill et al. (1972). For studies of the interaction between fields and Kerr black holes-studies performed using the above solution, and using analogous solutions to the electromagnetic and gravitational wave equations-see Bardeen, Press, and Teukolsky (1972), Misner (1972b), Teukolsky (1972), Ipser (1971), Press and Teukolsky (1973), and Chrzanowski and Misner (1973).]

*This box is based on Misner (1969a).
*The global structure of the Schwarzschild geometry, including the existence of two singularities and two exterior regions, was first discovered by Synge (1950). See Box 31.1.
*On the problem of Plateau see, e.g., Courant (1937), Darboux (1941), or p. 157 of Lipman Bers (1952).
$†$ The uncharged ( $Q = 0$ ) version was first found as a solution to Einstein's vacuum field equations by Kerr (1963). The charged generalization was first found as a solution to the Einstein-Maxwell field equations by Newman, Couch, Chinnapared, Exton, Prakash, and Torrence (1965). Only later was the connection to black holes discovered; see Box 33.1.
*This is not quite true. Kerr-Newman spacetime possesses, in addition to its two Killing vectors, also a "Killing tensor" which is closely linked to the Boyer-Lindquist coordinates $r$ and $θ$ . See Walker and Penrose (1970); also §33.5.
Extraction of energy from a
black hole by processes in
Extraction of energy from a
black hole by processes in the ergosphere

§30.1. WHY IS THE UNIVERSE SO HOMOGENEOUS AND ISOTROPIC?

§30.2. THE KASNER MODEL FOR AN ANISOTROPIC UNIVERSE

§30.3. ADIABATIC COOLING OF ANISOTROPY

§30.4. VISCOUS DISSIPATION OF ANISTROPY

§30.5. PARTICLE CREATION IN AN ANISOTROPIC UNIVERSE

§30.6. INHOMOGENEOUS COSMOLOGIES

§30.7. THE MIXMASTER UNIVERSE

Box 30.1 THE MIXMASTER COSMOLOGY

A Generalized Kasner Model

Introducing Space Curvature

Box 30.1 (continued)

Variational Principles

ADM Hamiltonian

Super-Hamiltonian

Box 30.1 (continued)

Mixmaster Dynamics

"Bounce" Interrupts Kasner-like Steps Toward the Singularity

Box 30.1 (continued)

Steady-State, Quasiperiodic Infinity of "Bounces" Approaching the Singularity

Summary

Speculations on Time and the Singularity

Box 30.1 (continued)

§30.8. HORIZONS AND THE ISOTROPY OF the microwave background

GRAVITATIONAL COLLAPSE AND BLACK HOLES

синотен 31

SCHWARZSCHILD GEOMETRY

§31.1. INEVITABILITY OF COLLAPSE FOR MASSIVE STARS

§31.2. THE NONSINGULARITY OF THE GRAVITATIONAL RADIUS

Box 31.1 THE "SCHWARZSCHILD SINGULARITY": HISTORICAL REMARKS

Exercise 31.1. TIDAL FORCES ON INFALLING EXPLORER

EXERCISE

§31.3. BEHAVIOR OF SCHWARZSCHILD COORDINATES AT r = 2 M r = 2 M r=2Mr=2 Mr=2M

§31.4. SEVERAL WELL-BEHAVED COORDINATE SYSTEMS

Figure 31.2.

A. EDDINGTON-FINKELSTEIN COORDINATES

Ingoing Eddington-Finkelstein Coordinates-Adopt r r rrr and V ~ V ~ widetilde(V)\widetilde{V}V~ as coordinates in place of r r rrr and t t ttt

Outgoing Eddington-Finkelstein Coordinates-Adopt r r rrr and U ~ U ~ widetilde(U)\widetilde{U}U~ as coordinates in place of r r rrr and t t ttt

Box 31.2 (continued)

B. TRANSITION FROM EDDINGTON-FINKELSTEIN TO KRUSKAL-SZEKERES

Box 31.2 (continued)

§31.5. RELATIONSHIP BETWEEN KRUSKAL-SZEKERES COORDINATES AND SCHWARZSCHILD COORDINATES

EXERCISES

Exercise 31.2. NONRADIAL LIGHT CONES

Exercise 31.3. THE CRACK OF DOOM

Exercise 31.4. HOW LONG TO LIVE?

Exercise 31.5. EDDINGTON-FINKELSTEIN AND KRUSKAL-SZEKERES COMPARED

Exercise 31.6. ANOTHER COORDINATE SYSTEM

§31.6. DYNAMICS OF THE SCHWARZSCHILD GEOMETRY

EXERCISES

Exercise 31.7. SCHWARZSCHILD METRIC IN ISOTROPIC COORDINATES

Exercise 31.8. REISSNER-NORDSTR ∅ ∅ O/\varnothing∅ M GEOMETRY

cmorest 32

GRAVITATIONAL COLLAPSE

§32.1. RELEVANCE OF SCHWARZSCHILD GEOMETRY

§32.2. BIRKHOFF'S THEOREM

EXERCISE

Exercise 32.1. UNIQUENESS OF REISSNER-NORDSTR ϕ ϕ phi\phiϕ M GEOMETRY [Track 2]

§32.3. EXTERIOR GEOMETRY OF A COLLAPSING STAR

EXERCISE

Exercise 32.2. REDSHIFTS DURING COLLAPSE

§32.4. COLLAPSE OF A STAR WITH UNIFORM DENSITY AND ZERO PRESSURE

Exercise 32.3. EMBEDDING DIAGRAMS AND PHOTON PROPAGATION FOR COLLAPSING STAR

Exercise 32.4. MATCH OF FRIEDMANN INTERIOR TO SCHWARZSCHILD EXTERIOR

Exercise 32.5. STARS THAT COLLAPSE FROM INFINITY

Box 32.1 AN IDEALIZED COLLAPSING STAR WITH FRIEDMANN INTERIOR AND SCHWARZSCHILD EXTERIOR

Initial State

Subsequent Evolution

Details of the Collapse

History of Collapse as Probed by Hypersurfaces ABCD:

Box 32.1 (continued)

§32.5. SPHERICALLY SYMMETRIC COLLAPSE WITH INTERNAL PRESSURE FORCES

Exercise 32.6. GENERAL SPHERICAL COLLAPSE: METRIC IN COMOVING COORDINATES

Exercise 32.7. ADIABATIC SPHERICAL COLLAPSE: EQUATIONS OF EVOLUTION [Misner and Sharp (1964)]

Exercise 32.8. ANALYTIC SOLUTIONS FOR PRESSURE-FREE COLLAPSE [Tolman (1934b); Datt (1938)]

Exercise 32.9. COLLAPSE WITH UNIFORM DENSITY

§32.6. THE FATE OF A MAN WHO FALLS INTO THE SINGULARITY AT r = 0 r = 0 r=0r=0r=0

§32.7. REALISTIC GRAVITATIONAL COLLAPSEAN OVERVIEW

EXERCISES

Exercise 32.10. PRICE'S THEOREM FOR A SCALAR FIELD Exercise 32.10. PRICE'S THEOREM FOR A SCALAR FIELD [See Price (1971, 1972a), also Thorne (1972), for more details than are presented here.]

A. Density Perturbations

§31.3. BEHAVIOR OF SCHWARZSCHILD COORDINATES AT $r = 2 M$

Ingoing Eddington-Finkelstein Coordinates-Adopt $r$ and $\tilde{V}$ as coordinates in place of $r$ and $t$

Outgoing Eddington-Finkelstein Coordinates-Adopt $r$ and $\tilde{U}$ as coordinates in place of $r$ and $t$

Exercise 31.8. REISSNER-NORDSTR $\emptyset$ M GEOMETRY

Exercise 32.1. UNIQUENESS OF REISSNER-NORDSTR $ϕ$ M GEOMETRY [Track 2]

§32.6. THE FATE OF A MAN WHO FALLS INTO THE SINGULARITY AT $r = 0$

Exercise 32.10. PRICE'S THEOREM FOR A SCALAR FIELD
Exercise 32.10. PRICE'S THEOREM FOR A SCALAR FIELD [See Price (1971, 1972a), also Thorne (1972), for more details than are presented here.]